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Estimating Intergenerational Returns to Medical Care: New Evidence from At-Risk Newborns\thanks{We thank Jason Lindo, Anastasia Terskaya, Jorge Rodr\'iguez, Atheen Venkataramani, and Roland Rathelot for thoughtful comments and conversations, and Jorge Fabres and Javier Cifuentes for information related to medical interventions and institutional details on neonatal health care practices.  We acknowledge insightful comments from various anonymous referees which have greatly improved this paper, and the editor Sarah Miller.  We are grateful to seminar audiences at the Barcelona GSE Summer Forum, IADB, LACEA, Pontificia Universidad Javeriana, University of Chile, University of Exeter, Universidad de la Plata, New School of Economics, and Universidad O'Higgins.  We thank Sofía Arredondo, Agustina Crozier, Francine Montecinos, and Daniel Pailañir for outstanding research assistance. Clarke acknowledges financial support from ANID Chile (FONDECYT Regular 1200634) and support from the ANID/Millennium Science Initiative Grant via the ``Millennium Institute for Market Imperfections and Public Policy (MIPP)''.  Replication materials for this paper are available at \href{XXXX-DOI-XXXX}{XXXX-DOI-XXXX}.}}
\author{Damian Clarke\thanks{Department of Economics, University of Exeter, Department of Economics, University of Chile, IZA, CAGE \& MIPP.  Address: Diagonal Paraguay 257, Santiago, Chile.  Contact email: \href{mailto:dclarke@fen.uchile.cl}{dclarke@fen.uchile.cl}.}
  \and Nicol\'as Lillo Bustos\thanks{Ministry of Economics, Chile.  Contact: \href{mailto:nlillob@economia.cl}{nlillob@economia.cl}.}
  \and Kathya Tapia-Schythe\thanks{Department of Economics, University of California, Davis.  Contact: \href{mailto:kattapia@ucdavis.edu}{kattapia@ucdavis.edu}.}
}
\date{\today}

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\begin{abstract}
We examine whether intensive early-life government-funded interventions targeted to at-risk newborns are transmitted intergenerationally.  Using a regression discontinuity design and administrative data from Chile we follow women up to the age of 25, and document the surprising fact that children of individuals who were treated at birth have worse indicators of health at birth a generation later.  We suggest this owes to selective fertility, finding that marginally treated individuals are substantially more likely to give birth.  These new stylised facts suggest that in certain circumstances, the long-term implications of public investments within family lineages may be quite different to their short-term implications, placing more weight on necessary reinforcing interventions.
\end{abstract}

\noindent \small{\textbf{JEL Codes}: I11; I18; J13; H51; O15.} \\
  \small{\textbf{Keywords}: Early life interventions; intergenerational mobility; parental investments; fertility; health care provision.} \\
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\section{Introduction}
Returns to early life public investment programmes accrue over life.  How generalised is this accrual, for how long does it last, and over what dimensions do investments accrue?  These questions have important implications for policy design related to optimal welfare investments over the life-cycle \citep{Cunhaetal2010,CurrieRossinSlater2015}.  Conceivably, if early-life circumstances shape later life outcomes, policies which affect such investments could have considerable intergenerational implications \citep{MogstadTorsvik2022}, suggesting impacts may echo well beyond the period of policy receipt. 

In this paper we seek to understand these questions, studying whether targeted and intensive publicly-funded medical  interventions in very low birth weight (VLBW) children immediately after birth are transmitted onto the outcomes of \emph{future} generations.   Using a multigenerational linked database of the universe of births in Chile and a regression discontinuity (RD) design, we examine whether women who were born weighing just below a specific (1,500 gram) treatment cut-off go on to have babies who have greater measurable stocks of health at birth a generation later.  We trace out the impacts of an individual's early life treatment on their future interactions with the health care system, the composition of their own family during childhood, their fertility behaviour upon maturity, and ultimately, the intergenerational transmission of intensive medical care receipt to measures of their children's health and well-being. 

Large public health programmes targeted to children with poor birth outcomes result in immediate improvements in health and survival \citep{Almondetal2010,Bharadwajetal2013,Chynetal2021}, which impact educational outcomes in childhood and adolescence \citep{Bharadwajetal2013,Daysaletal2022}, reduce future reliance on public social safety net programmes \citep{Chynetal2021} and positively affect other members of the family including siblings and mothers over both the short and long term  \citep{Daysaletal2022}.\footnote{These studies, based on a similar RD design as that used in this paper, point to VLBW cut-offs as significant determinants of well-being for individuals during early life and childhood. The study of \citet{Bharadwajetal2013} considers the first generation using the same data and the same context studied in this paper.  Their results document a strong and enduring `first-stage' to the longer term and intergenerational context we study in this paper. As we discuss in the methods and results section of this paper, we document that their earlier results still hold when working with a much extended sample and recent advances in optimal RD designs (as has been documented in other recent RD papers of this nature such as \citet{Daysaletal2022}).}  More generally, there is substantial evidence from a broad literature in economics documenting intergenerational transmission of health at birth \citep[for example]{CurrieMoretti2007,Nilsson2017,Giuntellaetal2019,LahtiPlkkinenetal2017,Royer2009}. At the same time, the way individuals interact with health policies, and the degree to which treatment receipt can shape future outcomes across the life course, is complex and multi-faceted.  Both \citet{CurrieRossinSlater2015} and \citet{Wust2022}, reviewing a broad range of studies, demonstrate that early life policies have appreciable impacts across multiple later life domains including  adult health, educational attainment, labour market outcomes, and other measures of socio-economic status.  If neonatal and early life health programmes lift up the life courses of their original recipients, and in turn spillover to future generations, this may suggest that the already large benefits of such programmes could be a (considerable) lower bound. However, if families or individuals change their behaviour as a downstream result of medical intervention (or lack of intervention), or if medical intervention shapes intermediate outcomes potentially shifting the life courses of individuals, the intergenerational returns to medical care may be entirely different to returns to the first generation. Indeed, intergenerational returns could even be negative if families make compensating investments in less treated individuals thus overwhelming initial investments received by more intensively treated individuals, or if medical receipt shapes future individual behaviours in other unexpected ways.\footnote{There is evidence, for example, that early life health shocks may interact with future events and investments, potentially magnifying the impact of initial shocks \citep{Duqueetal2019}. In general, family circumstance has large and pervasive intergenerational implications (see for example \citet{Heckmanetal2022}), which in the absence of some exogenous shock, may make pinning down the causal intergenerational impact of early life health care receipt difficult.}

In this study we trace out impacts of the receipt of intensive medical care very early in childhood on outcomes in childhood, adolescence, early adulthood, and spillovers into future generations.  Linking comprehensive microdata registries, we are able to observe all births in Chile between 1992-2018, their future inpatient hospitalisations, and their future fertility histories, and, for those women that go on to have \emph{their own} births, we observe the early life health stocks and survival records for their children.  Our results show that despite relatively large effects of the programme on infant mortality and days of hospitalisation in the first generation suggesting a substantive first generation impact \citep{Bharadwajetal2013}, there is clearly no positive intergenerational transmission to the health outcomes at birth of the \emph{children} of individuals who were treated at birth.  Indeed, more surprisingly, we observe quite clear negative transmissions of intensive medical treatment receipt at birth to the health outcomes of second generation of children, especially when studying outcomes very low in the distribution of health at birth, such as the likelihood that a child is born prematurely, or with a very low birth weight. 

We consider a number of channels which may explain the reversal of impacts across generations.  These include selection in terms of fertility, changes in fertility timing or the composition of individuals giving birth, survival selection upon treatment receipt in the first generation, and (first generation) parental reinforcing behaviours overwhelming initial positive effects of intensive early life health treatments.  We additionally consider the likelihood that the lack of observed effects owes to low statistical power. In conducting these tests, we find clear evidence of strong policy effects on fertility many years after treatment is received.  These results suggest a chain of policy impacts, whereby individuals who receive intensive medical treatment are more able to take births to term, with subsequent negative implications on average health stocks of affected cohorts of children.

While these findings are based on rich microdata, it is important to note that there are limits to these data.  Most clearly, we are only able to match mothers (and not fathers) with their children, and, given that high-quality birth records only exist in our (developing country) sample from 1992 onwards, we are limited with the maximum age-range of potential second generation mothers which we observe.  More specifically, we are limited to focus on women up to a maximum age of 25 years (see discussion in Appendix \ref{app:data}). Thus, while we work with the full sample of over 6.5 million births occurring in our study period, we view our findings as informative for young mothers, and do not make claims to broader external validity outside of this group.  Nevertheless, the results and stylized facts we document suggest revisions are required to the standard understanding of intergenerational transfers when policy can shape outcomes on both extensive (fertility) and intensive (health at birth) outcomes.  These results provide a counter-example, in this particular setting and sample, to the broad and growing literature which documents positive intergenerational transfers, at least if only intensive margin outcomes are observed. 

This result is novel, given that the overwhelming body of the literature suggests that there are positive intergenerational spillovers in health and in socioeconomic measures more broadly, and that such positive spillovers are observable in mean outcomes of the population. For example, in examining health shocks, \citet{Lee2014} points to intergenerational impacts still persistent in grandchildren due to their grandmother's exposure to a violent uprising while pregnant, with similar patterns observed when considering in utero declines in pollution exposure as a result of the US Clean Air Act \citep{ColmerVoorhies2020}. In a recent study \citet{Eastetal2023} document important positive intergenerational spillovers owing to Medicaid coverage in the prenatal period. More broadly, there is evidence to suggest that maternal exposure to stressful events in adolescence and adulthood is passed across generations \citep{Akreshetal2021}.  Similarly, \citet{Almondetal2012} document intergenerational transfer of exposure to disease across generations of mothers. \citet{BhalotraRawlings2011,BhalotraRawlings2013} document significant gradients in the exposure to health shocks of children based on the health of their mothers, additionally pointing to important channels of intergenerational transmission of health.  These first-generation exposures have also been shown to be reflected in later life health outcomes of second generation children \citep{Venkataramani2011,Bencsiketal2023}, to be observed in long-term measures such as life expectancy \citep{Blacketal2022}, to be pervasive across country income levels and settings \citep{Hallidayetal2021,Changetal2024,Halliday2020}, and to endure across more than two generations \citep{Costa2021}.  A rich stream of emerging literature has shown intergenerational transfers in education and well-being, with important implications on the accrual of inequality over generations.\footnote{Two studies have also considered the interaction of educational and health policies: \citet{RossinSlaterWust2020} examining nurse visits and pre-school, and \citet{BarrGibbs2022} who examine Head Start which includes health screenings, both finding substantial positive intergenerational effects.} 

This paper contributes to a range of literatures.  It joins a number of recent studies in documenting intergenerational transmission of early life health indicators.  It also contributes to a literature considering the intergenerational transfer of exposures to particular events or environments across cohorts.  It additionally joins a large literature considering the returns to public health programmes.  This is one of only a small number of studies to consider intergenerational implications of individual-level exposure to a large public health policy, allowing for the estimation of long-term returns of such programmes.\footnote{There are a number of studies which consider how exposure to policies can impact educational or labour market of exposed individuals lifting up trajectories of disadvantaged individuals, in this way breaking intergenerational cycles of disadvantage.  Examples of such policies, particularly in the Scandinavian context, are discussed by \citet{Wust2022}; see for example \citet{Butikoferetal2019}.  What is different in this paper is the consideration of how exposure to a health policy is transmitted into \emph{future} generations.} The study closest to ours in this regard is \citet{Eastetal2023}: an impressive multigenerational study of Medicaid, finding positive intergenerational spillovers in the US.  Importantly, our study and their study work with a broadly similar age range of second generation mothers (though of course a quite different exposure variable), suggesting a useful contrast to our proof of concept that \emph{in some settings} intergenerational spillovers can be negative when studying observed health outcomes. Perhaps the main contribution of this study, and where it expands knowledge in each of the literatures discussed above, is in documenting that the marginal returns to medical care in one generation can have considerable long-term implications, improving the outcomes of individuals in the first generation while at the same time bequeathing weaker health stocks to the following generation when a selective fertility channel is operating.\footnote{This paper also contributes, in a limited way, to a literature on the replicability crisis in social sciences, and concerns with the use of non-public data.  For a particular study \citep{Bharadwajetal2013} when previously private data was subsequently published without restrictions, we were able to substantively replicate the findings of the original paper's private data, starting from scratch in data collation and generation.  Further, updating results based on technical advances in the intervening period since the paper's publication, and additionally updating to include a substantially longer time-frame which more than duplicated the original sample points to results that are entirely consistent with those in the originally published research.} 

The structure of the paper is as follows. In Section \ref{scn:background} we discuss the context studied here and the nature of the VLBW assignment threshold as a determinant of early life medical care. Section \ref{scn:data} discusses the intergenerational linked administrative data generated for this study, and Section \ref{scn:modelMethods} discusses how this is used to estimate impacts of exposure based on an RD design, additionally presenting a simple model to understand the full content of RD estimates in this context.  Section \ref{scn:results} provides results as well as identification checks, discussing both mechanisms which could explain the observed results, and their implications on how we should conceive the (long-term) returns to medical care. Section \ref{scn:conclusion} briefly concludes.


\section{Background and Medical Care Regimes}
\label{scn:background}
Birth registration in Chile is universal, and the large majority of births (over 99\%) are attended in public hospitals or private clinics which follow national-level protocols set by the Ministry of Health. Births are overwhelmingly attended by doctors and/or midwives.  As laid out in \citet{Bharadwajetal2013}, national-level guidelines were set in 1991 by a national committee to standardise treatments at Neonatal Intensive Care Units (NICUs) in the country, which exist in each of Chile's 16 regions.  A particular concern at this point was the high rate of infant mortality among very low birth weight infants (weights below 1,500 grams).  Treatment protocols often explicitly mention 1,500 grams as a treatment cut-off, and these births undergo a systematic treatment protocol with follow-up procedures \citep{Hubneretal2009}.
  
\citet{Bharadwajetal2013}, pointing to \citet{Gonzalezetal2006,JimenezRomero2007,Palominoetal2005} note a number of explicit treatment cut-offs which are documented at 1,500 grams.  This includes the use of artificial lung surfactant, a complementary nutrition programme providing specialised supplementation (\emph{PNAC prematuro}), and a health care reform (\emph{AUGE}) defining neonatal follow-up appointments to screen for particular pathologies which are targeted at infants born at less than 1,500 grams.  These criteria are explicitly recognised also in later policy documents, such as the National neonatal guidebook issued to medical practitioners \citep{Menaetal2005},\footnote{Searches in this document suggest 24 occurrences of the use of a 1,500 grams, while other arbitrary cut-points are not similarly prevalent (e.g., 4 mentions of 2,000 grams and only 1 mention of 2,500 grams which is the cut-off for definitions of low birth weight (LBW)).} and when formally included as requirements for accessing policies (such as that indicated in the \emph{AUGE} reform), these require children to have weights \emph{strictly below} 1,500 grams, implying that those who weigh 1,500 grams will not classify for treatment.  

Discontinuous treatment assignments at 1,500 grams have been apparent in official clinical guidelines for an extended period of time.  The 2005 Neonatal Guide, referring to actions taken in the previous decade, notes policies targeted to reduce rates of morbidity among VLBW infants such as the national surfactant programme (and other programmes to standardise use of drugs such as Indometacin), programmes to follow infants over an extended period of time post-birth, the regionalisation of neonatal services \citep{Menaetal2005}, and recommendations that all newborns weighing less than 1,500 grams should receive diagnostic ultrasounds to examine intracranial haemorrhage, diagnostic tests for retinopathy, yearly specialised ophthalmological follow-ups, and a range of other specialised treatment courses in cases of particular diagnoses.  There is evidence that this discontinuity remains highly relevant even late in the period under study.  For example, the Clinical Manual for Neonatal Care written in 2016 for one of Santiago's large public hospitals with a high complexity neonatal care unit lists a large number of specific treatments which neonates should receive if they weigh under 1,500 grams.  These include specialised procedures for temperature maintenance at birth, a delayed clamping of the umbilical cord to allow for greater blood flow to the baby following birth, the required presence of two specialists trained in resuscitation, specific formulae for ventilation and nasal air flow masks, and direct transfer to neonatal intensive care units.  An entire chapter is dedicated to these procedures in the hospital's clinical guides \citep{MulhausenGonzalez2016}.  No similar such guidelines exist for other distributional points of birth weight. 


Many of these policies are additionally targeted at children who are born at less than 32 or 33 weeks of gestation, suggesting that the discontinuity may only bind for babies born at 32 weeks or above.  Indeed, this is an argument made by \citet{Bharadwajetal2013}, who note that often policy documents or formal selection criteria for programmes also note extreme prematurity along with very low birth weight.  While this is often the case, it appears that it is not uniformly so.  For example, in recent technical guidelines, a number of treatments are indicated as owing exclusively to the 1,500 gram threshold, and not gestational limits (e.g., temperature regulation procedures discussed in \citet{MulhausenGonzalez2016}).  As we discuss later in the paper, we also observe evidence suggesting elevated rates of hospitalisation for babies born just below 1,500 grams even among individuals born at 32 weeks or earlier.  Thus, while it is clear that treatment rules will bind most cleanly for babies born below 1,500 grams but at 32 weeks of gestation or greater, it is not necessarily clear that no discontinuity will exist around the 1,500 gram cut-off for babies born at below 32 weeks.  Thus while we generally follow \citet{Bharadwajetal2013} in focusing on births at above 32 weeks of gestation, we additionally show that results are robust to relaxing this sample restriction.

\begin{figure}
  \caption{\textbf{Intergenerational Transmission of Early Life Health Measures}}
  \label{fig:intergen}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/birthWeightIntergen.pdf}
    \caption{Maternal birth weight and children's birth weight}
    \label{fig:intergenBW}
  \end{subfigure}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/lbwIntergen.pdf}
    \caption{Maternal birth weight and child's low birth weight status}
    \label{fig:intergenLBW}
  \end{subfigure}
  \floatfoot{Notes: Each sub-plot documents average outcomes of individuals born to mothers whose birth weight is indicated on the horizontal axis.  Panel (a) considers the birth weight of second-generation children, while panel (b) considers the average proportion of second-generation children who are low birth weight (weight $<$ 2500 grams). Optimally spaced bins and their confidence intervals are documented as black points and error bands.  A cubic B-spline and its 99\% confidence interval is overlaid as a continual solid line and shaded area.  Optimal definitions and recommendations follow \citet{Cattaneoetal2019}.}
\end{figure}
  
 
In considering intergenerational links in health at birth, a stylised fact is that there is considerable, but not complete, closure of gaps in health at birth across generations.  In Figure \ref{fig:intergen} we document, using microdata on all intergenerational links between births in Chile occurring after 1992, that while there is a clear tendency of lower birth weight children to go on to have births which themselves have poorer average health outcomes at birth, the intergenerational links are considerably tempered, with all binned averages located above 3,000 grams.  These binned scatter plots suggest that on average, individuals weighing as little as 1,500 grams have children with birth weights of over 3,000 grams, with this value rising steadily, reaching about 3,400 grams on average for individuals who weighed 4,000 grams at birth.  Thus, a considerable reversion to the mean is observed, suggesting the existence of some intergenerational dependence in health measures at birth, but with considerably cushioned impacts.  Similar patterns are observed when considering low birth weight indicators, with children of LBW mothers being more likely to themselves be LBW, but nevertheless, 90\% of individuals whose mothers were LBW ``graduate out'' of this status in the next generation.  This result lines up with a broader literature focused on intergenerational transmission of health in adulthood (\textit{i.e.}\ between parents and their adult children). Intergenerational health associations of 0.2-0.3 are typically found when considering both physical and mental health across a range of settings \citep{Halliday2020,Hallidayetal2021,Changetal2024}.   The results observed for birth weight here consistent with the lower end of this range, suggesting intergenerational associations of around 0.2.\footnote{This can be appreciated in Figure \ref{fig:intergen}(a): when shifting from mother's weight of 2500 to 4000g, average children's weight shifts from around 3150 to 3500g, an association of 0.21.}  

\section{Data}
\label{scn:data}
We generate matched microdata covering all the 6,617,637 births occurring in Chile between 1992 and 2018.  These births are matched to their future survival history, inpatient hospitalisation records, and any of their own births occurring in the future.  In the case of any future births, we observe the birth outcomes, and survival history of their children.  For each birth we observe both child- and parent-level measures.  Thus, these data are longitudinal covering up to 3 generations: characteristics of the mother and father of children born in generation 1, as well as characteristics of any births which occur later to children of women from generation 1.  These generation 2 births allow us to observe the characteristics of babies from generation 1 when they go on to have their own children.  In total, of the 6,617,637 births occurring between 1992-2018, 6,147,623 have valid information for all relevant variables and meet maternal age criteria discussed below.  Of those births, 3,010,251 (48.97\%) are girls, 420,389 of whom go on to have their own future live birth. For these births we thus observe their outcomes at birth and in early life, their mother's outcomes at birth, early life and at the date of their child's birth, and their grandparents' characteristics at the moment of their child birth. 

We follow \citet{LuVogl2022} in referring to an intergenerational link as a lineage.  We observe lineages which consist of mothers and their children, where mothers are observed both when they were born, in which case information on their mother (the ``grandmother'') is observed, and if they give birth, are additionally observed at this time, in which case information on their children is observed. We lay out this nomenclature, and the structure of lineages in our data in Appendix Figure \ref{fig:lineage}. Birth certificate records which are provided by the Ministry of Health contain each mother's national identification number, but do not contain the father's national identification number.  For this reason, we are only able to work with links between a \emph{mother's} health across generations as matching boys to their children is impossible.  Available evidence suggests that father's health is also transmitted to future generations \citep{Giuntellaetal2019}, though to a lesser degree than mothers \citep{Changetal2024}, which is a point we return to when considering total policy effects in Section \ref{sscn:resultgen2}.

\paragraph{Birth, Death and Hospitalisation Data}
Birth registries in Chile are universal, estimated to cover 99.9\% of all births.  Birth registries contain high quality records of birth weight (in grams), gestational length (in weeks), and size at birth (in cm), as well as information on the place of birth, and mother's and father's education and employment, along with other covariates.\footnote{These data do not, however, contain information related to factors such as BMI, weight gain during pregnancy, smoking, or other behavioural measures.}
Individuals are recorded using their national identity number, assigned at birth. Data on these births are merged (using a masked version of the national identity number) with the hospitalisation registry and the death registry, which cover all deaths and in-patient hospitalisations in the country.  In total, 58,973 births are matched to a death record before the age of 1 year.  And in total 83,841 births are observed to appear in the death registry at any point during this period.  


For a number of mechanism tests, we consider hospitalisation records. Prior to 2001, individual identifiers are missing from a considerable proportion of micro-level registries of hospitalisations (inpatient records).  Given this, we do not consider hospitalisation data prior to 2001, instead consistently working with subsets of birth cohorts for whom hospitalisation records are complete (for similar definitions see \citet{MenaresMunoz2025}). Thus, when we examine impacts of early-life health investments on later life health care usage, we work with age-specific outcomes.  For example, when considering hospitalisation at age 1, we can examine this for cohorts whose hospitalisations are observed completely at this age, namely individuals born from 2001--2017.  And when considering hospitalisation at age 2, we work with the sample of births from 2000-2016, and so forth for other ages.  Of all births, 2,924,795 are matched to at least one hospitalisation, and in total 5,656,409 hospitalisations are matched with births, implying that the average number of hospitalisations per matched birth is 1.93.  These hospitalisations cover all inpatient care provided in the country, both in public hospitals and private clinics, and include information on both the reason for hospitalisation (recorded by standardised ICD-10 codes), as well as the duration of hospitalisation in nights. 

\paragraph{Additional Data}
In a number of cases, we consider parental and family responses to child birth. In these cases, we generate information on changes following observed births, namely whether families go on to have additional births and if so the birth spacing in months, whether children are observed to be covered by private, rather than public, insurance or are treated in private, rather than public, hospitals, and whether parents are observed to leave or join the labour market, or move to a higher wage industry.\footnote{Birth registries contain information on each individual's occupation.  We cross each individual's occupation with information on the average salary and average hours worked in each occupation by region from large household surveys conducted every 2-3 years (Chile's CASEN survey), thus allowing us to measure average conditions in the industry in which mothers and fathers work.} In each case, these measures are generated from administrative registers, and so are subset to individuals which appear in the relevant registers.  For example, in the case of private insurance and hospitalisation, this measure is generated only for children who are observed to be hospitalised.  Similarly, in the case of birth spacing and labour market changes, these measures are observed only for individuals whose parents go on to have a subsequent birth, as both of these measures are generated based off information contained in birth registries.

\begin{table}[ht]
  \caption{\textbf{Summary Statistics -- All Births}}
  \label{tab:sumstatsFull}
  \scalebox{0.92}{
  \begin{tabular}{lccccc} \toprule
    & Obs.\ & Mean & Std.\ Dev.\ & Min.\  & Max. \\ \midrule
    \multicolumn{1}{l}{\textbf{Panel A: First Generation Births}} &&&&& \\
    \input{tables/Summary_G1}
    \multicolumn{1}{l}{{\textbf{Panel B: Second Generation Births \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}} &&&&& \\ 
    \input{tables/Summary_G2}
    \bottomrule
    \multicolumn{6}{p{17.4cm}}{\footnotesize Notes: Summary statistics are displayed for all births occurring in the first generation (births from 1992-2018), as well as births matched to prior births (second generation births).  The full sample consists of all births occurring between 1992 and 2018 in Chile from administrative data maintained by the Ministry of Health, and here we focus on mothers aged 15-45 at time of birth as used in principal estimation samples. Certain observations do not have information for certain variables based on exposure definitions: for example death within 1 year of birth is only defined for individuals aged at least 1 year, and number of hospitalisation days is only observed if individuals are hospitalised.}
  \end{tabular}}
\end{table}

\paragraph{Sample Matches} Sample matches are generated based on an individual's national identity number (the RUT) which is unique, ubiquitous, assigned at birth and used throughout life.  In Appendix Table \ref{tab:matches} we document the matches across registers.  As noted above, not all births are matched with hospitalisations given that many individuals are never hospitalised, and not all hospitalisations are matched to births, as many hospitalisations occur to older individuals who were not born during the period of 1992--2018.  Of particular interest is the cross within the birth registry, identifying the universe of all lineages where mothers were born from 1992 onwards.  These 420,389 lineages by definition must only occur to mothers aged at most 26 years (mothers who were born in 1992 and had a birth in 2018).  We document the full matrix of mother-child birth years in Appendix Table \ref{tab:birthChart}, which makes clear the composition of our intergenerational sample.  Because for certain tests we consider individuals up to specific ages, we remove the small portion of individuals who have turned 26 years, as no observation in our sample has completed their 26\textsuperscript{th} year of life. While this is a restricted sample, a feature of this sample is that it is somewhat similar in age range to that of \citet{Eastetal2023}, who observe women up to the age of 28 when considering intergenerational links.  This allows for some comparability of our results with theirs based upon women's age. In the case of Chile, in the period under study this is approximately the median age (refer to Appendix Figure \ref{fig:Agemothers}). It is, however, important to note that this is a limitation in terms of data coverage, and in later years, we will observe younger women than our maximum age considered of 25.  Figure \ref{fig:Agemothers_d} plots the age distribution of our intergenerational sample, and we discuss this limitation in Appendix \ref{sscn:limits}.  In Section \ref{sscn:fertTiming} we return to discuss the implications of this sample for the internal and external validity of our findings. 

\paragraph{Summary Statistics}
Summary statistics for the matched microdata of the first and second generations are provided in Table \ref{tab:sumstatsFull}. We observe average birth weights of around 3.3 kg among the full sample, and slightly lower when considering second generation births.  These second generation births will necessarily be born to younger mothers (with an average age of 19.6 years, compared with 27.2 years in the full sample).  A low proportion of births are VLBW (around 1\% of the full sample), and on average most pregnancies are taken to full term (around 38.6 weeks in both the first and second generation sample).  In Table \ref{tab:sumstatsClose} we document identical summary statistics, but condition only on individuals born close to the 1,500 gram treatment cut-off.  In this case we observe a much greater proportion of VLBW infants (given the sample definition), very premature infants, and deaths within 1 year of birth.  In general, mothers in this sample are slightly older, at 28 years in the full sample.


\section{Model and Methods}
\label{scn:modelMethods}
\subsection{Justifying a Regression Discontinuity Design for Early Life Investments}
\label{sscn:estimation}
As in \citet{Bharadwajetal2013}, consider the observed birth weight of individual $i$, denoted $BW_i$, which imperfectly proxies unobserved health at birth, $H_i$: \vspace{-3mm}
\[
BW_i = H_i + \varepsilon_i.
\]
Additionally, consider the inputs received by an infant at hospital, $D_i$, which depend, decreasingly, on health at birth, as well as discontinuous treatment assignment rules:
\[
D_i = g(H_i) + \kappa \cdot  \mathbb{1}\{BW_i<c\} + \nu_i,
\]
specifically, here neonatal health treatments are shifted upwards by some amount $\kappa$ when an individual is born with a birth weight below the cut-off $c$.
In this setting, initial medical care $D_i$ is correlated with unobserved health measures not captured by birth weight, implying that estimates of the impacts of early life health investments conditional on birth weight will be biased in standard models. This leads to the RDD:
\begin{equation}
  \label{eqn:baseline}
  y_i =  f(BW_i-c) + \alpha\cdot \mathbb{1}(BW_i<c)+X_i\beta + \upsilon_i,
\end{equation}    
where $f(\cdot)$ captures local relationships between the running variable (birth weight) and outcomes of interest $y_i$, specifically allowing for split local-linear or higher order polynomials, $X_i$ is a vector of covariates, and $\upsilon_i$ an unobservable error term. This design allows for the impact of medical treatments to be isolated from unobserved health stocks in the neighbourhood of $c$, given that crossing the threshold assignment causes a discrete jump in treatment $\Delta D_i=\kappa$, provided standard RD assumptions related to continuity of unobservables surrounding the cut-off point are met.

\paragraph{Estimation and Inference Procedures}
We estimate \eqref{eqn:baseline}, with the baseline specification following quite standard procedures.  Namely, our principal model consists of a split local-linear parametrisation for $f(BW_i<1500)$ within MSE-optimal bandwidths following \citet{Calonicoetal2020a}, a triangular kernel, and robust bias-corrected inference of \citet{Calonicoetal2014}.  We follow \citet{Bharadwajetal2013} in defining specific details of the principal specification, diverging from this only where clearly-justified arguments exist, such as in the use of optimal bandwidth selection procedures, as these procedures have been largely developed following the publication of their paper.  Thus, we control for covariates indicated by \citet{Bharadwajetal2013} which are maternal characteristics (education, age, marital status), type of birth service (midwife or doctor), birth region, and year of birth. We additionally include a heaping control at 50 gram intervals, a point we turn to discuss below.  While we aim to follow the models by \citet{Bharadwajetal2013} as baseline specifications, we also document the stability of estimates to a range of alternative modelling considerations including alternative kernel weightings, polynomial orderings, and the use of donut RDDs \citep{Barrecaetal2011}.

\paragraph{Validity of this RDD}
This RDD has been previously discussed and validated in a number of settings, and we conduct a suite of tests laid out in this literature.  These are discussed at more length in Appendix \ref{app:RDvalidity}, and include tests for balance of predetermined observable factors around the cut-off, tests of heaping of the running variable following \citet{Almondetal2010,Bharadwajetal2013,Barrecaetal2011}, tests for effects at placebo cut-offs, and tests for manipulation of the running variable.  While our interest in this study is on the intergenerational impacts of early life interventions, we also document the initial relevance of these treatments in impacting infant mortality.  These results are all discussed in Section \ref{sscn:robustness} and Appendix \ref{app:RDvalidity}.

\paragraph{Principal Outcomes and Multiple Inference}
As we wish to consider how the impacts of early life health receipt accrue over life, and into future generations, we necessarily conduct multiple hypothesis tests considering multiple outcomes and a single treatment receipt.  To avoid concerns that any findings will simply owe to inflated type I errors from repeated hypothesis testing, we proceed in two ways.  Firstly, across all principal intergenerational measures we generate a single outcome index, following \citet{Anderson2008}.  Secondly, within all classes of outcomes considered, we report both standard p-values, and q-sharpened p-values, which control for the false discovery rate. 



\subsection{What do reduced form estimates capture?  A Conceptual Model}
\label{sscn:model}
Understanding what $\alpha$ captures depends upon how posterior events and investments interact with $D_i$.   \citet{Bharadwajetal2013,Chynetal2021} discuss this in a single generation setting (see Appendix \ref{sapp:singleGenModel} where this is applied to our notation). We extend this into a multigenerational setting, where there are clear implications of selection into the second generation which may interact with initial treatment.

When we consider how early life treatments can have intergenerational consequences, this requires explicitly taking into account individual selection into appearing in the second generation. Specifically, we must consider that a first generation woman's fertility decisions may depend on her own initial inputs at birth, as well as the way these interact with future outcomes.  Naturally, an individual will only appear in the second generation if a first generation woman has given birth.  Thus, consider each woman's fertility decisions modelled at early ages (say her teenage years) and older ages (her 20s), which is described in terms of a series of latent variables as follows:
\begin{eqnarray}
  \label{eqn:selec1a}
  \text{Early Fertility}^{*}_i &=& \gamma_1^{E} H_i + \gamma_2^{E} D_i + \gamma_3^E I^{post}(H_i,D_i) + \gamma_4^E B^{post}(H_i,D_i) + \upsilon^E \\ 
  \label{eqn:selec1b}
  \text{Later Fertility}^{*}_i &=& \gamma_1^{L} H_i + \gamma_2^{L} D_i + \gamma_3^L I^{post}(H_i,D_i) + \gamma_4^L B^{post}(H_i,D_i) + \upsilon^L.
\end{eqnarray}
Individuals for whom $\text{Early Fertility}^{*}_i>0$ will have $\text{Early Fertility}_i=1$, otherwise, $\text{Early Fertility}_i=0$, with an identical crossing rule for $\text{Late Fertility}^{*}_i$.  This is a simple model of fertility behaviour in which fertility at a given age is a function of an individual's health at birth $H_i$, treatment receipt at birth $D_i$, as well as total parental investment received after birth $I^{post}(\cdot)$, and individual behaviours after birth $B^{post}(\cdot)$.  These latter two elements may both interact with initial health and medical treatment receipt.\footnote{While this is a simplified model it provides a framework for understanding policy mechanisms, as discussed below.  We document in Appendix \ref{app:model} that it can be extended in multiple ways, which we will reference when introducing specific modelling elements.  In Appendix \ref{sapp:ferttime} we discuss modelling fertility at each age, and in Appendix \ref{app:dynamicHealth} note that health can be viewed as a dynamic stock without greatly altering implications for our setting.} 



In turn, we may posit that  birth weight of individual $i$'s child (the second generation birth, who will be indexed $j$) can be described in the following way:
\begin{equation}
  \label{eqn:selec2}
  BW_{ij} = H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(H_i,D_i)+ \varPsi B^{post}(H_i,D_i)+ X_{it}\beta_t + \nu_{ij},
\end{equation}
which is observed only if a generation 1 mother gives birth.  Note that this function now depends on heritable factors, and so the mother's early life health, $H_i$, health interventions at birth $D_i$, as well as posterior investments in the mother, $I^{post}(\cdot)$, and behaviours of the mother, $B^{post}(\cdot)$, all may explain her child's birth weight. In \eqref{eqn:selec2} birth weight ($BW$) is used to fix ideas, but this is considered representative of the observed proxies of health at birth we discuss in Section \ref{scn:data}. 

Our interest in this paper is to determine the impact of intensive treatment at birth in the first generation on health outcomes of the following generation.  Thus, our dependent variable of interest measures human capital at birth.  To consider what an RD estimate captures where the outcome is birth weight of the second generation, we can write the expected birth weight of the second generation \emph{conditional} on the child appearing in the second generation in terms of the well-known \citet{Heckman1974} selection equation.  If we assume joint normality of each of $\upsilon^E$ and $\upsilon^L$ with $\nu_{ij}$ with arbitrary covariance terms $\rho_E$ and $\rho_L$ respectively, then, combining equations \eqref{eqn:selec1a} and \eqref{eqn:selec1b} with \eqref{eqn:selec2} we have:
\begin{eqnarray}
  \label{eqn:HeckmanA}
  E[BW_{ij}|\text{Early fertility}_i=1]&=& H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(\cdot) + \varPsi B^{post}(\cdot) \nonumber \\ &&+  \rho_{E}\sigma\lambda_{E}\left[H_i,D_i,I^{post}(\cdot),B^{post}(\cdot)\right]  \\
  \label{eqn:HeckmanB}
  E[BW_{ij}|\text{Later fertility}_i=1]&=& H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(\cdot) + \varPsi B^{post}(\cdot) \nonumber \\ &&+ \rho_{L}\sigma\lambda_{L}\left[H_i,D_i,I^{post}(\cdot),B^{post}(\cdot)\right].
\end{eqnarray}
Here for simplicity we have written $I^{post}(H_i,D_i)$ as $I^{post}(\cdot)$, and similarly for $B^{post}$.  A key point to note here is that selection is now captured by the (age-specific) inverse Mills Ratio $\lambda$, the standard deviation of $\nu_{ij}$, and (age-specific) covariance term $\rho$. The assumption of joint normality is convenient in allowing a parametrisation of birth weight in terms of easily-understandable components in \eqref{eqn:HeckmanA}-\eqref{eqn:HeckmanB} although not required for our results below.  We discuss a control function approach in Appendix \ref{sapp:noHeckmanNormal}. 

These selection equations \eqref{eqn:HeckmanA}-\eqref{eqn:HeckmanB} have clear implications for the RD estimate in the case where outcomes capture indicators of health at birth of the second generation.  While the regression discontinuity design levies locally exogenous changes in treatment for first generation individuals, the total policy-relevant treatment effect on the second generation will consist of multiple posterior interactions with the policy.  Specifically, noting from \eqref{eqn:baseline} that $\alpha\equiv\frac{\partial y}{\partial \mathbb{1}(BW_i<c)}$, and given the local smoothness assumption in RD which states that nothing varies around the cut-off except for treatment receipt, then $\alpha\equiv\frac{\partial y}{\partial D}\kappa$.  Thus, based on the model laid-out above, the coefficient of interest when considering second-generation outcomes such as birth-weight can be written as:
\begin{center}
\begin{tabular}{lll}
$\widehat\alpha\quad =$ &  $\psi\kappa$ & ``Structural''\\
 &  $+\quad \varphi\cdot\Delta I^{post}(c) + \varPsi\cdot\Delta B^{post}(c)$  & ``Behavioural''\\
 &  $+\quad \rho_{E}\sigma\Delta\lambda_{E}\left[c,\Delta I^{post}(c),\Delta B^{post}(c)\right] \qquad\qquad$  & ``Selection (Early Fertility)''\\
& $+\quad \rho_{L}\sigma\Delta\lambda_{L}\left[c,\Delta I^{post}(c),\Delta B^{post}(c)\right]$ & ``Selection (Later Fertility)''\\
\end{tabular}
\end{center}
where $\Delta X(c)$ generically refers to any change in element $X$ occurring when crossing the cut-off at $c$.

Based on this model, the reduced form impact of a mother $i$ of child $j$ crossing the 1,500 gram threshold is decomposed into five terms. First, $\psi\cdot\kappa$ denotes the direct intergenerational transmission of improved health at birth of the mother, on to her children (the `structural effect' of the policy). Second and third, $\varphi\cdot\Delta I_t^{post}(c)$ denotes the spillover on subsequent generations of parental investments in mother $i$ which are sensitive to the policy and $\varPsi\cdot\Delta B^{post}(c)$ refers to individual behaviours which are sensitive to policy receipt. This (grand-)parental investment channel will capture any changes in first generation parenting as a result of treatment-receipt which are then transmitted to the second generation, and the (parental-)behaviour channel captures any individual responses to the policy which may impact children.  Finally, the fourth and fifth composite terms capture selection into the second generation, or a fertility channel of the policy which is allowed to be time-sensitive.  This term, $\rho_E\sigma \Delta \lambda_E[c,\Delta I^{post}(c),\Delta B^{post}(c)]$ makes clear that this selection may operate in a number of ways.  Given that the inverse Mills Ratio in equation \eqref{eqn:HeckmanA} contains $D_i$ as well as $I^{post}(D_i)$ and $B^{post}(D_i)$, selection may owe to direct policy impacts (e.g. healthier individuals as a result of intervention at birth may be more likely to take pregnancy to term), and/or selection may owe to changes in parental investments (e.g. compensating behaviour by individuals may increase the likelihood of fertility).  In both cases, we are concerned only with the way which treatment receipt changes this selection, as if fertility is unchanged as a result of the policy $\Delta\lambda=0$, and the fourth and fifth selection terms will be null.  Note that, importantly, if fertility selection is relevant, the scale term $\rho_E$ capturing unobserved correlations between birth weight and fertility likelihood will impact both the sign and magnitude of the selection term, underscoring the importance of understanding selection into fertility.  

\paragraph*{Implications}
This simple model has a number of clear implications, which we take forward in studying the mechanisms of effects documented in Section \ref{scn:results}.  One clear implication is that our RDD estimate includes multiple channels, such that the total effect of treatment need not be the direct effect of intergenerational transmission of health investments in observable stocks of health at birth (the `structural' channel above).   And the second implication is that the estimated quantity $\widehat\alpha$ can diverge from this structural effect if any of a number of channels are observed to be relevant.  

The first channel is behavioural.  If parental or individual behaviour changes when crossing the treatment threshold, this will potentially explain treatment effects provided that such behaviours affect health at birth.  A second channel is selection into fertility.  If we observe changes in rates of childbirth when crossing the treatment threshold, this will potentially explain treatment effects provided that unobserved elements of selection into treatment and birth weight are correlated.  A third channel refers to timing of childbirth.   Treatment receipt may shift individuals from later to earlier childbearing (or vice versa) even if it does not affect their total fertility, and this can potentially explain treatment effects provided that unobserved elements of selection into treatment and birth weight differ among younger and older women.  In the results section we first present estimates of $\widehat\alpha$, before considering each of these theoretical channels as a guide for our analysis of the mechanisms which explain $\widehat\alpha$.  We finally comment on what one can infer about the direct structural channel when any of the other channels of intergenerational transmission are found to be present.


\section{Results}
\label{scn:results}
\subsection{Intergenerational Impacts of Health Intervention at Birth} \label{sscn:resultgen2}
\subsubsection{Principal Results} 
\label{sscn:resultgen2main}
In Figure \ref{fig:RDPlotGen2} we present visual RD plots, and in Table \ref{tab:Gen2BirthOutcomes} we present formal tests for intergenerational transmission of intensive health treatments at birth.  Coefficients capture the impact of a mother having received intensive treatment when she was born on her children's health when she gives birth many years later.  Figure \ref{fig:RDPlotGen2} plots average outcomes in 20 gram birth weight bins within \citeauthor{Calonicoetal2014}'s Mean Square Error-optimal bandwidth around the discontinuity, overlaid on quadratic fits and their confidence intervals.  Table \ref{tab:Gen2BirthOutcomes} presents robust, bias-corrected estimates based on local linear regressions with a triangular kernel.  The stability of these estimates to alternative functional forms, bandwidth and modelling choices is discussed in Section \ref{sscn:robustness} of the paper.

Across all outcomes considered, there is no evidence to suggest positive intergenerational gradients. For example, in the case of birth weight, there is a clear \emph{upward} shift when crossing from below the cut-off (more treated) to above the cut-off (less treated) individuals in Figure \ref{fig:f2BWRDD}, and the point estimate in Table \ref{tab:Gen2BirthOutcomes} shows a return of $-217$ grams on the next generation.  This is a strong negative effect, significant at the 5\% level if considering uncorrected p-values, or at the 10\% level if consider the FDR corrected p-value (0.065). However, some of these estimates are imprecisely estimated. While there are a large number of intergenerational links covered in these 30 years of data (more than 400,000 for the 32+ week samples), when focusing on the optimal bandwidths this is reduced to a small number effective observations.\footnote{Power is a challenge in this setting.  As we lay out in Appendix Figure \ref{fig:PowerGen2}.}
Nevertheless, despite noisy estimates, in all measures observed in Panel A point estimates are consistent with there actually being no, or negative transmission of early-life investments, rather than positive returns.  As well as point estimates and standard errors (presented in parentheses in the table), we additionally present the p-value on a one sided test that corresponds to these early life investments having negative intergenerational impacts.  Thus, a low p-value would point to evidence to reject the null that the effect of these investments are negative, and high p-values can be considered as providing very little evidence to reject this null.  Across all outcomes considered, these p-values range from 0.716 to 0.984, suggesting very little evidence to reject the null of a negative effect in favour of the alternative of a positive impact. 
 
\begin{landscape}
\begin{figure}[htpb!]
  \caption{\textbf{Descriptive Plots of Parental Policy Receipt and Child Health Measures}}
  \label{fig:RDPlotGen2}  
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_PESO.eps}
    \caption{Birth weight}
    \label{fig:f2BWRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_SEMANAS.eps}
    \caption{Gestational period}
  \label{fig:gestRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_TALLA.eps}
    \caption{Gestational length}
    \label{fig:sizeRDD}
\end{subfigure}

\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_vlbw.eps}
    \caption{Very low birth weight}
    \label{fig:VLBWRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_premature.eps}
    \caption{Prematurity}
  \label{fig:premRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/RDplot_fgrate.eps}
    \caption{Fetal growth rate}
    \label{fig:sgaRDD}
\end{subfigure}
\floatfoot{Notes: Plots show separate quadratic fits estimated on each side of the 1,500 gram cut-off, in each case restricting attention to observations within the optimal bandwidth following \citet{Calonicoetal2014}.  95\% confidence intervals of the quadratic fit are estimated, and circles represent average outcomes in 20 gram bins.  The size of each point reflects the relative number of observations in each bin. Observations at 1,500 grams are plotted in binned averages, but are not used in estimating the quadratic fit.}
\end{figure}
\end{landscape}



Point estimates in the case of gestational weeks point to an insignificant reduction by around 0.6 weeks, and in the case of birth length an insignificant reduction of around 0.3 cm. Likewise, in the case of infant mortality of the second generation, we observe an insignificant effect, however the sign is consistent with individuals who receive intensive treatments going on to have children who are less likely to survive.  In Panel B we consider a number of alternative derived measures based on variables recorded in the birth register.  In the case of prematurity and low birth weight, these are critical measures of health stocks at points quite low in the distribution of birth weight and gestational weeks.  In this case, at low points in the health distribution we find considerable negative returns to a mother's early life health receipt.  In the case of both prematurity and low birth weight, we observe the individuals whose mothers just classified for treatment receipt are more than twice as likely to suffer from these conditions as individuals whose mothers were marginally above the treatment threshold.  We similarly observe a statistically significant reduction in fetal growth rate (defined as weight divided by gestational weeks).  The final outcome in Panel B is an index of health measures at birth which is used to reduce the dimension of the test to a single dimension, and avoid inflated type I error rates.  Here, in line with results observed throughout the table, this child health index is observed to be considerably worse among children of parents who received early life treatment, at around 0.42 of a standard deviation \emph{lower} on average.


\begin{table}[ht!]
  \caption{\textbf{Intensive Health Investments and Birth Outcomes of the Second Generation}}
  \label{tab:Gen2BirthOutcomes}
  \scalebox{0.99}{
  \begin{tabular}{lcccc} \toprule
    & Gestation  & Birth weight 
    & Birth length & Infant   \\ 
    & (weeks) & (grams) & (cms) & Mortality \\
    \textbf{Panel A: Baseline Variables} & (1) & (2) & (3) & (4) \\
    \midrule
    \input{tables/T2A_o32}
    \midrule
    & Prematurity & Very low  
    & Fetal growth  & Anderson \\
    & & birth weight & rate & Index \\
    \textbf{Panel B: Transformed Measures} & (5) & (6) & (7) & (8) \\
    \midrule
    \input{tables/T2B_o32}
    \bottomrule
    \multicolumn{5}{p{14.8cm}}{{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for mothers. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}.  Robust bias corrected standard errors are reported in parentheses.  Below standard errors, a one tailed t-test is calculated, which can be viewed as the support in favour of there actually being \emph{positive} intergenerational transmission to the second generation. q-sharpened p-values refer to corrections conducted across the entire class of outcomes. 
    * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
  \end{tabular}}
\end{table}
 
The results from Table \ref{tab:Gen2BirthOutcomes} suggest that while tests on mean outcomes point to negative but imprecise results, particular key points of the distribution of these outcomes do see substantial shifts as a result of the receipt of intensive early-life investments.  This is particularly the case in low birth weight measures, where we observe that a mother's receipt of intensive investments at birth makes her far \emph{more} likely to have low birth weight children.  Given the high costs, both in terms of individual well-being as well as hospital-level costs for outcomes very low in the health distribution \citep{Almondetal2010}, we consider these distributional effects in a more flexible setting in Figure \ref{fig:distGen2}.  In this case, each coefficient and confidence interval refers to the probability that a mother has a birth with health measures \emph{below} particular cut-offs indicated on the horizontal axis.  Each coefficient and CI is thus generated from its own RD estimation, and is interpreted as the impact of policy receipt on intergenerational transfer of poor health indicators. 

Figure \ref{fig:bwDist} shows clear negative impacts of maternal policy receipt on her child's health.  In proportional terms, this is particularly relevant in the far left tail of the distribution of birth weight.  For example, in the case of 1,500 grams, we estimate that a mother who receives intensive early life health treatment increases her probability of having a birth at less that 1,500 grams by 4.5pp, which is an effective tripling considering that the proportion of such births occurring to mothers in this bandwidth of interest are around 0.015.  Similar such patterns are observed when considering weights up to 2,250 grams.  It is important to note here that this implies that (costly) early-life interventions in generation 1 imply a much greater proportion of births having to receive such interventions in generation 2.  In the case of gestational length documented in Figure \ref{fig:weeksDist}, similar such distributional impacts are seen, with consistently positive probabilities of observing short gestational lengths, significant in the case of 31 weeks (extreme prematurity), with particularly large impacts observed at 37 weeks (prematurity).  Again, these results are consistent with \emph{negative} intergenerational gradients of health intervention at birth, and certainly never suggestive of positive gradients as frequently observed in prior literature.

\begin{figure}[t]
  \caption{\textbf{Distributional Impacts of Early Life Health Interventions on Second Generation Health Stocks}}
  \label{fig:distGen2}  
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/DistBW_o32.pdf}
    \caption{Birth weight}
    \label{fig:bwDist}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/DistWeeks_o32.pdf}
    \caption{Gestational length}
    \label{fig:weeksDist}
  \end{subfigure}
\floatfoot{Notes: Each point estimate (black square) and 90 and 95\% confidence interval (dark and light shaded areas respectively) refer to RD estimates of the likelihood that a birth occurring to a treated girl born has health stocks (birth or gestational weight) below the cut-off indicated on the x-axis of each sub-plot.  Each estimate is generated using the same sample and methods described in Notes to Table \ref{tab:Gen2BirthOutcomes}.}
\end{figure}

\subsubsection{Robustness}
\label{sscn:robustness}
The previous results are based on optimal RD procedures following standard choices, such as the use of bias correction, local linear estimation procedures, and weighting with a triangular kernel centred at the cut-off point.  To avoid concerns related to specification search, we have precisely followed \citet{Bharadwajetal2013} in the use of control variables.  Nonetheless, we document the robustness of principal results to alternative specifications or empirical decisions. We summarise here tests presented in Appendix \ref{app:robust} which suggest these results are robust to alternative reasonable choices.
 
In Appendix Figures \ref{fig:BandwithVar32plus}-\ref{fig:BiasBandwithConstant} we document that results are broadly consistent across reasonable bandwidth choices, and evolve as expected as bandwidths grow considerably. We consider this in two ways: firstly (in Figure \ref{fig:BandwithVar32plus}) we simply vary the bandwidth of data in which the RD model is fitted, using this same bandwidth for bias-correction, and secondly (in Figure \ref{fig:BiasBandwithConstant}) we increase the bandwidth, maintaining constant the ratio between the bandwidth used for estimation and bias-correction.  Generally speaking, across intergenerational outcomes, results are observed to be largest when focusing on a bandwidth more tightly bounded to the cut-off, and grow smaller only when the bandwidth is pushed up considerably, to as much as 300 grams.  This value is well above optimal, and reassuring in that identification is local, and to the degree that wider bandwidths are used, more bias is expected.  

More generally, we consider a range of alternative models varying, firstly the bandwidth selection and/or use of bias correction, and secondly the functional form of the running variable.  These results are provided in Appendix Tables \ref{tab:altspec1}-\ref{tab:altspec2}.  These consider a number of procedures for each intergenerational measure considered.  First, we report robust-bias corrected results (replicating results from the main text).  Second, we provide results using an alternative manner of selecting the optimal bandwidth which has been shown to be optimal to minimize rates of error in hypothesis testing  \citep{Calonicoetal2020a}. Third, we present results using `standard' RD models with conventional variance estimates, and fourth we present bias-corrected RD results, again using conventional variance estimates.  Each of these results are presented for split linear and split quadratic polynomials to capture the relationship between outcomes and the running variable. Across models, results point to consistently negative intergenerational relationships (in the case of birth weight, prematurity, fetal growth rate, VLBW status and the \citet{Anderson2008} index), or negative but insignificant results in the case of gestational weeks and birth length.  One outcome (infant mortality) suggests noisier results, with signs flipping in certain cases, suggesting that this outcome should be considered with some caution, given its rarity in second generation births.  Results are also robust to estimating a Donut RD with a range of donut hole radius values (Appendix Table \ref{tab:robustness}), and do not in general appear to be driven by jumps owing to any heaping at rounded birth weights, with null effects largely observed when considering placebo designs at birth weights of 1250, 1750, 2000, 2250 and 2500 grams.

Finally, as we noted in Section \ref{scn:background}, our estimation sample consists of all individuals born at weeks 32 and above, who are unambiguously exposed to the policy.  We additionally present main results for the full sample of both individuals born at 32 weeks and above, as well as individuals born below 32 weeks.  In general, we expect results may be slightly attenuated, given that it is less clear that these individuals are exposed to all aspects of the policy.  These results are provided in Table \ref{tab:Gen2BirthOutcomesALL} and Figures \ref{fig:RDPlotGen2ALL}-\ref{fig:distGen2All}.  Results are quantitatively similar, though slightly dampened in certain outcomes.

\subsection{Understanding channels of intergenerational effects}
\label{sscn:channels}
How can we rationalise the fact that we observe evidence which suggests \emph{negative} transmission of early life health interventions across generations when the broad consensus from extant literature  is that these early life policies have broadly positive impacts on \emph{first generation} individuals,\footnote{It is worth noting that in a cost-benefit sense, the positive first generation impacts in terms of lives saved far outweigh the costs of the programme, even when accounting for long term costs (see Appendix \ref{sscn:discussion}).} and positive shocks are generally observed to transmit across generations?  It seems exceedingly unlikely that the direct effect of intergenerational transfers (the `structural channel' laid out in Section \ref{sscn:model}) is negative, which points to some other relevant countervailing channel. The conceptual model laid out in Section \ref{sscn:model} suggests three potential competing explanations, and we use this model to guide an analysis of these channels here.


\subsubsection{Channel 1: Behavioural Responses to Early-life Treatment}
A first channel which may explain these results owes to ways in which individuals and families react to treatment receipt.  As highlighted in Section \ref{sscn:model} parental investments may react to treatment receipt, or individual or family behaviour may otherwise change as a result of being exposed to treatment.  


\paragraph{Compensatory or Reinforcing Behaviour}
A key channel which may explain null or negative intergenerational transmission of policy impacts relates to parental compensatory behaviour.  As laid out in the model in Section \ref{sscn:model}, any change in parental behaviour as a \emph{response} to treatment receipt at birth may act to counteract or reinforce any direct policy impacts.  Specifically, a channel may exist in which parents whose children are born marginally above the 1,500 gram threshold and hence who are observed to be relatively worse off early in life, invest more heavily in these children, compensating initial disadvantage, and indeed fully closing the gap, explaining null or negative policy impacts in the long-run.  This is something which can be empirically tested, if in place of examining the specific early life health measures in generation 2 births, we examine as outcomes in a RDD specification parental investment behaviours across the life of their children.

To do this we collect a number of measures of parental investments in their children, or parental behaviours which may impact children's outcomes or well being.  These are classified in terms of health investments, future demographic decisions, and parental labour market sorting decisions.  In the case of health investments, we examine whether, conditional on being hospitalised, children are covered by (more expensive) private insurance schemes, or treated in private hospitals, which implies higher average out of pocket spending \citep{Crispietal2020}.  In terms of demographic decisions, we examine whether treatment receipt impacts future parental fertility, either changing the number of future births or their spacing.  In terms of labour market sorting, we test to see if \emph{following on from their child birth}, mothers or fathers are observed to join or leave the labour market, to switch to a higher paying employment sector, or to switch to a less hour-intensive employment sector. Such labour market responses could potentially impact child well-being on various margins: first an income margin if one or both parents opts to join to the labour market to afford greater investments in children, or second, a time investment if parents are observed to shift into less `greedy' careers (as defined in \citet{Goldin2021}), in favour of increasing (time) investments in children.  Evidence of such labour movements and desires to seek careers which support both labour market and family investments are discussed, for example, in \citet{GoldinKatz2016}.


\begin{table}[ht]
  \caption{{\textbf{Parental Responses to Treatment Receipt}}}
  \label{tab:parentLabour}
  \scalebox{0.8}{
  \begin{tabular}{lcccccccc} \toprule
    \multicolumn{9}{l}{\textbf{Panel A: Labour Market Responses}}\\
    &\multicolumn{4}{c}{Mothers} & \multicolumn{4}{c}{Fathers} \\ \cmidrule(r){2-5}\cmidrule(r){6-9}
    & Leaves  & Joins  & Expected      & Expected & Leaves  & Joins  & Expected      & Expected \\
    & Market  & Market & Hour change &  Salary change & Market  & Market & Hour change & Salary change \\ \midrule
    \input{tables/T3A_o32_hps_edit}
    \midrule
    \multicolumn{9}{l}{}\\
    \multicolumn{9}{l}{\textbf{Panel B: Parental Health Investments and Fertility Behaviour}}\\
    &\multicolumn{3}{c}{{Private Insurance}} & \multicolumn{3}{c}{{Private Hospitalisation}} & Future & Birth \\ \cmidrule(r){2-4}\cmidrule(r){5-7}
    & 2 & 4 & 6 & 2  & 4  & 6 & Birth & Spacing \\ \midrule
    \input{tables/T3B_o32_hps_edit}
    \bottomrule
    \multicolumn{9}{p{20.4cm}}{{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for newborns. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}. Robust bias corrected standard errors clustered at the gram level are reported in parentheses. p-values for one-sided tests are shown in square ($H_1$: negative) brackets. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}}
\end{table}

We lay out tests of parental responses to treatment receipt in Table \ref{tab:parentLabour} Panel A (parental labour market responses) and Panel B (changes in health investments or future fertility behaviour).  In Panel A, RD estimates suggest relatively little evidence of affected mothers changing intensive margin labour supply decisions (leaving or joining conditional on their previous labour market status), however we do observe evidence consistent with mothers of treated children moving into higher wage sectors or participating in sectors which experienced higher wage growth.  This is observed in column 4, with mothers of treated children being observed to change to sectors with mean monthly salaries which are approximately 80,000 CLP higher (around 80-85 USD based on exchange rates in 2025).  In the case of fathers, we observe relatively little evidence of similar changes in labour market circumstance.  The one exception to this is an increase in the likelihood that fathers of marginally treated children join the labour market conditional on previously having been out of the labour market, though we note that this effect is driven off a very small sample of fathers who previously had not participated in the labour market, and so is considerably underpowered. In general, this result suggesting larger labour market responses for mothers than for fathers in the face of child health shocks is consistent with findings in the broader literature, for example \citet{Eriksenetal2021} who find maternal shifts in labour market choices following negative child health shocks, and a broad literature, lead by \citet{Klevenetal2019}, which documents relative inflexibility of father's labour supply to child birth, a phenomenon also noted in Chile \citep{Berniell2021}. While these parental labour market responses to early life investments are of interest in their own right, for our results here, if anything they suggest that the observed patterns cannot be explained by labour market results, as mothers of individuals born just below the 1,500 gram threshold are observed to move to industries with higher, rather than lower salaries, without being much greedier in terms of time demands.


In the case of parental health investments and fertility responses (Table \ref{tab:parentLabour}, Panel B), we observe relatively little evidence suggestive of consistent changes in the way which parents invest in their children's health care.  This coheres with evidence presented by \citet{Bharadwajetal2013} who found relatively little evidence of changes in educational investments by parents of marginally treated versus marginally untreated children.  Early in life, we observe that conditional on hospitalisation, rates of private hospitalisation and private insurance coverage did not significantly differ across the 1,500 gram threshold.  Similarly, we observe no evidence to suggest that parents of treated individuals were more likely to go on to have another birth, or change the timing of future births.  In fact, both birth timing and the number of future births following a very low birth weight baby are quite similar to outcomes in the general population (Appendix Figure \ref{fig:birthSpace}).

In general, these results suggest that (first generation) parental responses can explain relatively little of the observed negative intergenerational impacts to children of the second generation.  Had we observed clear evidence of compensating investments, where individuals just to the right of the cut-off received additional investments or otherwise more positive home environments, this may have suggested that initial favourable medical treatments of treated individuals were overwhelmed by later favourable investments in untreated individuals. If anything, we observe that the reverse may be true, given the relatively better labour market trajectories for mothers of treated children.


\paragraph{Behavioural Responses to the Policy: Interaction with the Health System}
One potential explanation for these results is that a mother's stock of health at birth is passed on to her baby. Specifically, given the positive relationship between maternal health stocks and child health at birth \citep{Lassietal2013,CurrieCole1993}, for a health channel to explain \emph{negative} intergenerational transmission, we would require that mothers who were marginally treated by the policy at birth be \emph{less} healthy than mothers who were marginally untreated.  While unlikely, such a phenomena could occur, if, for example, individuals who were less treated at birth were more likely to be hospitalised in early life, and ended up accruing more positive health stocks by maturity.   Thus, the key consideration for this mechanism is whether health stocks appear to be different, and in particular, whether there is evidence of compensatory investments in health during an individual's fertile period.

\begin{figure}[h!]
  \caption{\textbf{Long-Term Health Stocks and Early Life Interventions}}
  \label{fig:RDhealth}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/hospitalization_o32.eps}
    \caption{Hospital days}
    \label{fig:fertRDHosp}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/hospitalizationNoExt_o32.eps}
    \caption{Hospital days, `chronic' causes}
    \label{fig:fertRDHospChronic}
    \end{subfigure}
    \floatfoot{Notes: Each point estimate and confidence interval refer to the impacts of early life medical investment on an individual's days of hospitalisation (panel (a)), and days of hospitalisation related to chronic conditions (panel (b)).  Thicker black error bars present 90\% CIs, while thinner error bars report 95\% CIs.  Days of hospitalisation (in both cases) are measured as totals for all individuals who have reached the age indicated, and take the value of 0 if the individual is not hospitalised this year, or otherwise a positive integer reporting the total number of days spent in hospital.  All estimates follow the procedures laid out in Section \ref{scn:modelMethods}, and report RBC estimates using a local linear regression with a triangular kernel in the MSE optimal bandwidth. 
    }
\end{figure}


In Figure \ref{fig:RDhealth} we examine the number of days an individual is hospitalised by year, estimated following the RD models laid out above at each age from 0 up until 24.  Figure \ref{fig:fertRDHosp} estimates the impact on the total number of days that affected individuals spend in the hospital for all causes, and panel (b) estimates the impact of early life medical treatment on hospital days which are classified as chronic.\footnote{Chronic hospitalisations are classified using the Chronic Condition Indicator (CCI) developed by the Healthcare Cost and Utilization Project (HCUP).} In both panels it is apparent that there are relatively small or non-existent impacts on the long term usage of health care.  Both when considering all hospital days, and hospital days for chronic causes, after the first few years of life there is little evidence of an enduring effect on hospitalisations.

We \emph{do} observe a clear policy impact on hospitalisations early in life, estimating an increase of around 4 days in the year which birth occurs, 0.5-1.5 days per year up to year 3, and then smaller and non-significant impacts there-after.  These effects are in line with those documented in \citet{Bharadwajetal2013}, reaffirming the power of the policy in terms of health investments.  However, the lack of clear later life results suggests that a role of direct transmission of health stocks from mother to children may be limited, given that we do not observe that individuals who marginally qualified for treatment have considerably different health stocks at maturity, at least if health stocks are proxied by hospitalisations. 


\paragraph{Behavioural Responses to Treatment: Education, Partnership, Family Spillovers and Migration}
There are a range of alternative possible explanations which we are not simple to test with the available data, but which can be illuminated by examining the available literature or aggregate statistics.  This includes educational attainment and school peer interactions or relationship formation more generally, within-family spillovers, and selective out-migration. 

\paragraph{Education} Impacts of early life medical receipt on education have been documented in a number of settings \citep{Bharadwajetal2013,Daysaletal2022}, and so may partially explain observed results.  \citet{Bharadwajetal2013} documents that educational outcomes up to grade 8 improved on average among individuals who marginally received treatment, while \citet{Daysaletal2022} find improvements in education in ninth grade test scores, though no significant effect on the likelihood to enrol in higher education.  While our data is fully matched within the health system, we are not able to observe educational attainment of individuals who do not give birth, and so cannot directly test whether education is observed to increase in this setting and age group more generally.  However we are able to observe education at the age when mother's give birth, and in Table \ref{tab:condEducPartners} document that results are found to hold when conditioning on both mothers and grandmother's education at birth, as well as when only focusing on individuals from highly educated families (as proxied by grandmother's education, given that young mothers are potentially still completing their education).  We find results are broadly consistent when controlling for education, and when conditioning on high education (and indeed, are slightly larger in this group).  Because education is, if anything, likely to be higher among individuals marginally below the 1,500 gram threshold, it seems unlikely that educational attainment itself can explain worse outcomes among individuals who received treatment.  However, we return to explore this point in more detail in Sections \ref{scn:selectionAnalysis}-\ref{sscn:fertTiming}, where we consider whether educational attainment may interact with any selection into child birth, or timing of child birth.

\paragraph{Partnership}
A second explanation relates to individual interaction with treatment and the marriage market.  One possibility is that individuals who are exposed to early life treatment form better partnerships, with impacts on support during pregnancy and outcomes at birth.  Once again, as in the case with education, this seems unlikely to drive results in a direct fashion, and in Table \ref{tab:condEducPartners} we show that results are robust to controlling for a proxy of partner presence as well as considering only individuals with observed partners.  As in the case with education, we are not able to observe partnership characteristics of women who do not give birth, and so return to discuss this point in Section \ref{scn:selectionAnalysis}-\ref{sscn:fertTiming} when considering whether this interacts with selection into second generation birth or birth timing.

\paragraph{Alternative Behavioural Explanations: Within-family spillovers or Migration}
There are precedents suggesting that early life health interventions at this margin can result in positive within-family spillovers, improving mother's mental health as well as sibling outcomes \citep{Daysaletal2022}.  While our data does not allow us to consider such measures in this context, for this to explain a \emph{negative} intergenerational health spillover, these positive family spillovers such as improved maternal mental health would need to map negatively into health of the following generation, which seems unlikely.  One final explanation related to selection owes to migration out of the country.  Such migration would need to be selective on one side of the treatment threshold.  This explanation also seems unlikely given that emigration rates in Chile are very low, at 0.35 per 1,000 individuals, and generally concentrated at working ages, particularly from 35 onward \citep{Stefoni2011}.

\subsubsection{Channel 2: Selection Into Fertility}
\label{scn:selectionAnalysis}
A second theoretical channel which may explain negative intergenerational effects is treatment-mediated \emph{selection into fertility}.  If as a result of treatment individuals give birth when they would not have given birth without treatment, this could potentially explain the results documented in Section \ref{sscn:resultgen2}.   Indeed, it is also true that selection could originate even at the time of birth of the first generation given that the treatment impacts survival of (potential) parents when they are born.  While here our focus is on selective fertility, we discuss briefly first that survival selection when originally receiving treatment is unlikely to drive our results.

\paragraph{(A) Survival selection in generation 1}
While the main analysis focuses on selection into second-generation births through differential fertility, a related concern is that the policy may also have induced selection through survival at birth in the first generation, given evidence that intensive neonatal treatment increased survival among infants with poorer initial health stocks. To assess whether such selective survival could account for the negative intergenerational effects we observe, we conduct counterfactual exercises that impute the fertility and health outcomes of individuals who would have survived absent the policy.  This is discussed at more length in Appendix \ref{app:2dim} (Table \ref{tab:counterfactuals} and Figure \ref{fig:select2d}). Across a wide range of scenarios—varying assumed health at birth across the observed distribution and imputing fertility profiles based on birth-weight-specific averages—selective survival can attenuate the magnitude of estimates but is generally insufficient to reverse their sign. Even under extreme assumptions, such as highly fertile counterfactual individuals giving birth to children at very low health percentiles, negative intergenerational effects persist for most outcomes, including birth weight, gestational length, and fetal growth. Only for size at birth do more plausible counterfactuals partially offset the estimates. Overall, these results suggest that selective survival at birth in the first generation alone is not sufficient to explain away the observed negative intergenerational transmission across health dimensions.

\paragraph{(B) Selection into fertility later in life}
Figure \ref{fig:fertDescHealth} presents descriptive plots of the likelihood that a woman gives birth by specific ages, graphing probabilities by an individual's own weight at birth.  These plots are based on all women exposed to the possibility of giving birth---that is individuals who have reached the age under consideration in each plot, and hence are potential `second generation' mothers.  Plotted values represent the actual proportion of these women who actually do become mothers.  Each point refers to the average in 50 gram bins, with the size of the point representing the number of individuals on which the average is based.  Any `selection' of interest would be apparent by changes in rates of birth just at the point where treatment targeting ends.

\begin{figure}[ht!]
\caption{\textbf{Fertility and Stocks of Health at Birth}}
\label{fig:fertDescHealth}
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.9\linewidth]{figures/childBy16.eps}
    \caption{Child by Age 16}
    \label{fig:fert16}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.9\linewidth]{figures/childBy19.eps}
  \caption{Child by Age 19}
  \label{fig:fert18}
\end{subfigure}

\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.9\linewidth]{figures/childBy22.eps}
  \caption{Child by Age 22}
  \label{fig:fert22}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.9\linewidth]{figures/childBy25.eps}
  \caption{Child by Age 25}
  \label{fig:fert25}
\end{subfigure}
\vspace{-4mm}
\floatfoot{Notes: Each figure plots the likelihood that a women born at a particular birth weight goes on to give birth by the age indicated in the plot caption.  Points represent average proportions in 50 gram bins based on the women's weight when she was born.  Averages are calculated from the full sample of women observed in the birth register, who have reached the age indicated in each plot.  The size of each point refers to the relative number of women in this birth weight bin.  The red dashed line indicates the 1,500 gram VLBW threshold used to assign discontinuous medical intervention at birth.}
\end{figure}


Prior to considering selection \emph{per se}, it is immediately notable and noteworthy that there is a steep gradient in the likelihood of giving birth by each of the ages documented in Figure \ref{fig:fertDescHealth} at lower points of the birth weight threshold.  For each of the ages documented, there is a steep gradient in rates of birth up to around 2,500 grams.\footnote{For ease of  visualization, only 4 plots are displayed in the main text.  In a previous working paper version of this study \citep{Clarkeetal2022} we have documented patterns for each age.} For example, in the case of births by the age of 19, individuals who survive to age 19 and were born below 1,000 grams have less than a 10\% chance of having given birth, rising to around 12\% among individuals between 1,500-2,000 grams, before levelling off at 15\% above around 2,500 grams.  While points low in the birth weight distribution are based on few observations, the regularity of this pattern, both within and across age groups is clear, with this gradient consistently being observed, and consistently flattening out from above around 2,500 grams.

As far as we are aware, this stylised fact has not been previously documented in the economic literature, and the few medical studies which have documented correlations between birth weight and fertility are observational and based on relatively small convenience samples \citep{vikstrom2014,vanderpal2021,vangendt2015}, or indeed consider other birth outcomes such as preterm birth \citep{Swamy2008}. These results suggest considerable returns to birth weight in ways not previously considered in the economic literature.  It also interacts with a broader literature on labour market returns to health at birth, and women's labour market returns in particular.  If higher birth weight individuals have higher educational and labour market returns \citep{BehrmanRosenzweig2018,Bharadwajetal2018}, and at the same time birth weight is positively correlated with fertility, then negative links between fertility and labour market outcomes \citep{Addaetal2017,Bloometal2009} may partially obscure the full human capital returns to birth weight.


In Figure \ref{fig:fertDescHealth} it is additionally notable to the plain eye that there is an important discontinuity in rates of birth which are observed around the 1,500 gram cut-off.  Even in using quite crude 50 gram bins in the spirit of capturing descriptive patterns, in each case clear sharp declines are observed in rates of birth when moving from just below 1,500 grams (marginally treated individuals) to just above 1,500 grams (marginally untreated individuals).  For example, in Figure \ref{fig:fert22}, around 25 percent of individuals are observed to have a birth before the age of 22 in the 50 gram bin just below 1,500 grams, with this proportion falling to around 20 percent, or declining by approximately one quarter, when moving to the bin just above 1,500 grams.  These values are identified formally using identical RDD methods as used throughout the paper in Figure \ref{fig:fertAbortRDD}.  Here age-specific estimates are presented, estimated following equation \ref{eqn:baseline}, where the outcome considered is the total number of births a (potential) second generation mother has by age $x$ (panel (a)), or if she has had a birth by age $x$ (panel (b)).  We observe large, and generally significant, impacts across all ages considered.  By the age of 22, estimated impacts described in Figure \ref{fig:fertRDD} show that the number of births for individuals receiving the policy has increased by 0.1, against a base of approximately 0.28 births in individuals in the estimation bandwidth.  Similarly large estimates are observed at each age above 20, growing to 0.2 additional births by the age of 25, versus a mean of around 0.48. These values are age-specific and grow by age, although if we estimate a model pooling across all ages, this suggests that fertility increases by a statistically significant 0.048 births, versus a mean of 0.214 births (Table \ref{tab:fertBirths}).

\begin{figure}[h!]
  \caption{\textbf{Impacts of Early Life Health Interventions on Fertility and Spontaneous Abortions}}
  \label{fig:fertAbortRDD}
   \begin{subfigure}{.45\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/fertility_o32.eps}
    \caption{Fertility}
    \label{fig:fertRDD}
    \end{subfigure}
    \begin{subfigure}{.45\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/fertility2_o32.eps}
    \caption{Individual Has Given Birth}
    \label{fig:fertbinRDD}
    \end{subfigure}
    
   \begin{subfigure}{.45\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/abortions_o32.eps}
    \caption{Abortions}
    \label{fig:abortRDD}
    \end{subfigure}
   \begin{subfigure}{.45\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{figures/sexratio_o32.eps}
    \caption{Male Child}
    \label{fig:maleRDD}
    \end{subfigure}
    \vspace{-4mm}
 \floatfoot{Notes: Each point estimate and confidence interval refer to the impacts of treatment on the number of births an individual has had by each age (panel (a)), the probability an individual has had a birth by each age (panel(b)), the number of abortions observed in hospitalisation data  (panel (c)), and the likelihood that an observed birth is a male (panel (d)).  Estimates in panel (a)-(c) are defined for all second generation individuals surviving to each indicated age, and so each estimation sample consists of all individuals.  Estimates in panel (d) are limited to individuals who have had a birth by the age indicated on the x-axis, and so are limited to births occurring at (at least) age 18, as otherwise samples are too small to reliably estimate RDD models.  Error bars 90\% and 95\% CIs.  All estimates follow the procedures laid out in Section \ref{scn:modelMethods}. Results for all women including those born below 32 weeks are provided in Figure \ref{fig:fertAbortRDDALL}.}
\end{figure}

This result suggests that policy receipt has a clear long-run impact on treated girls.  It makes them considerably more likely to give birth, as much as a quarter of a century after the initial policy receipt.  This also provides a key potential explanation of the unexpected \emph{negative} transmission of health at birth into the following generation.  The policy may have a positive effect on fertility by rescuing marginal births of treated individuals.  Individuals who in the absence of the policy would not have conceived or not have given live birth to a baby, do give birth to a baby when receiving intensive treatment at birth.  This finding is consistent with the policy saving (second generation) babies which are marginally weaker, providing one explanation of the negative impacts observed in Section \ref{sscn:resultgen2} of this paper.  We return to discuss evidence in favour of this interpretation in Section \ref{sscn:marginalVsAverage}.

Fertility selection is thus a key mechanism which can potentially explain the negative transmission across generations.  We briefly consider what explains this fertility result (the `mechanism of the mechanism'), prior to turning to potential alternative selection channels.  Higher rates of childbirth following a shock such as policy receipt may occur in a number of ways.  Firstly, it may be the case that individuals are equally likely to conceive, but just less likely to take births to term when they do not receive the intensive early life medical investment.  Or secondly, it may be that the early life medical intervention provides individuals greater social or reproductive resources to conceive.  Due to the nature of administrative records and high rates of miscarriage, we cannot observe all conceptions occurring around the cut-off, but rather only actual birth rates.  As a proxy of miscarriage, or births not taken to term after conception, we can observe all spontaneous abortions which result in hospitalisations.\footnote{These are observed in inpatient records, and are inferred from ICD-10 codes which are recorded as the reason for treatment.  A full tabular classification is provided in an earlier version of this working paper \citep{Clarkeetal2022}.}  This is likely a considerable lower bound of all conceptions not taken term.  We examine the impact of policy receipt on the likelihood of suffering a spontaneous abortion in Figure \ref{fig:abortRDD}.  Analysis is conducted identically to the analysis of fertility. Here it is appears that based on \textit{this} proxy, higher rates of birth among treated individuals do not owe to a greater likelihood of taking births to term, but rather, more likely due to a greater propensity to conceive.  


However, an alternative and potentially more sensitive proxy of selective survival in utero is the sex ratio of births.  It is well known that male fetuses are less likely to survive to birth, and are more sensitive to shocks in utero \citep[see, \textit{e.g.}][]{Valente2015}.  Thus, an improvement in rates of survival of males at birth is evidence of improvements in rates of survival among relatively weaker babies \citep{TriversWillard1973}.  We replicate our analysis from panels (a) to (c) with the likelihood that a child is male in panel (d). Based on this measure, we \textit{do} see evidence in favour of fertility results being driven by selective survival of weaker fetuses in exposed mothers.  In Figure \ref{fig:maleRDD} we observe substantial improvements in rates of birth of male children on the policy-exposed side of the RD threshold, and this becomes particularly noteworthy at ages where fertility increases most.  These effects are large, suggesting around a 10pp increase in the likelihood of male children being observed among live births.\footnote{This is apparent even among descriptive statistics, with the likelihood of a birth being male in the 250 grams below the threshold being 0.47, compared with 0.52 in the 250 grams above the threshold.} This suggests the observed results can, at least in part, be ascribed to birth selection occurring after conception, and potentially relatively early in gestation given a lack of clear impacts on spontaneous abortion.  It is also important to note that this explanation coheres with the distributional results we observed in Section  \ref{sscn:resultgen2}.  If treatment receipt saves second generation births which would not have been taken to term without the policy, we may expect increases in births with very poor health outcomes, which is what we did in fact observe with indicators of very low birth weight.

\begin{landscape}
\begin{figure}
  \caption{\textbf{Is There Policy-Driven Selection into Childbirth?}}
  \label{scn:selection}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_ageMum.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_educMum.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_employedMum.eps}
    \end{subfigure}   

    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_ageDad.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_educDad.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_employedDad.eps}
    \end{subfigure}   
    
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_marrieda.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_difedad.eps}
    \end{subfigure} 
        \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/figure7_RDplot_urbano.eps}
    \end{subfigure} 
\floatfoot{Notes: Each plot documents characteristics of mothers, fathers, or births of second generation births surrounding the 1,500 gram treatment application threshold of first generation births.  A quadratic polynomial fit is graphed on either side of the cut-off, and points reflect average outcomes in 20 gram birth weight bins, with relative sizes reflecting the number of individuals in each birth weight bin.  Each plot is displayed within \citet{Calonicoetal2020a}'s MSE optimal bandwidths.  Formal tests of discontinuities at 1,500 grams are provided in Appendix Table \ref{tab:fertilitySelectionRDD}.}
\end{figure}
\end{landscape}


This mechanism is also able to rationalise the effect sizes documented when considering second generation outcomes at birth.  Consider for example the impact on rates of VLBW in Table \ref{tab:Gen2BirthOutcomes}, which is an increase of 4.5pp, compared to a mean rate of 1.5pp among unexposed individuals. In essence, what this means is that the proportion of VLBW births jumps from 1.5\% to 6\% when crossing the treatment threshold.  Note also that we find that intensive medical treatment increases births by around 20.4\% (0.048/0.235 from Table \ref{tab:fertBirths}), and these infra-marginal births are those births which would have not survived without treatment receipt.  It is most likely that a very high proportion of babies that do not survive are very low birth weight (and generally have very poor health stocks).  However, even if we take a reasonably conservative view and assume that only around one quarter of these births which did not survive without policy intervention would have been very low birth weight, this can justify our effect.\footnote{This is calculated by dividing the 347 births in the below threshold (treated group) in Table \ref{tab:Gen2BirthOutcomes} into infra-marginal births (20.4\%, or 71 births) and always-survivors (79.6\%, or 276 births).  Given that on average 6\% of surviving births are VLBW, we can calculate the rate of VLBW among infra-marginal births as: $(0.06\times 347-276\times 0.015)/71=0.23$.}  The fact that we find a quite large effect on fertility, and that infra-marginal births will naturally be those with poor health stocks, suggests that our effect sizes are credible, and smaller effect sizes are unlikely unless infra-marginal births have unexpectedly high measures of health at birth.

In Figure \ref{scn:selection}, we examine whether this fertility selection is a generic result in the population, or something which is driven by particular groups.  Formal RD estimates which correspond to each plot in Figure \ref{scn:selection} are provided in Appendix Table \ref{tab:fertilitySelectionRDD}.  To test for selective fertility, we examine characteristics of mothers and fathers who go on to give birth in generation 2 around the 1,500 gram birth weight threshold.  This figure is thus the intergenerational analogue of typical balance tests conducted in RDD to examine whether observations on each side of the cut-off are balanced, or are selected in some way.  However, here, rather than acting as an identification check, these allow us to test for selective entry into the second generation of mothers.  Across 9 outcomes observed in administrative data (mother's and father's education, age and employment, marital status, and whether births are multiple), we observe some evidence consistent with policy-driven fertility changes being more prevalent among certain groups.  For example, we observe weak evidence to suggest that less educated women are slightly over-represented among treated rather than control individuals, which would be consistent with marginal births being more likely to survive among less educated women when they receive the treatment, compared with when they do not.  We observe more clear evidence to suggest that treated individuals have \emph{more} preferable partner matches when partners are observed: fathers are on average closer in ages to mothers, more likely to be employed, and mothers and fathers are more likely to be married.  Such a result may be consistent with treated individuals more generally forming better partnership matches as a result of treatment receipt.  In the following sub-section we consider some suggestive tests related to whether compositional changes in parents can explain our observed results. 

\subsubsection{Channel 3: Fertility Timing}
\label{sscn:fertTiming}
The relevance of selection into fertility implies that it is important to consider birth timing.  This is important for two reasons.  A first is related to the internal validity of these estimates.  As we consider mothers up to the age of (at most) 25, if treatment receipt makes mothers more likely to give birth early in their life, but then they do not give birth later, our estimates may simply be capturing a mechanical change in health at birth given that (over the age horizon we consider), children born to younger women have poorer stocks of health.\footnote{While results are necessarily noisier when stratifying by age, suggestive results presented in Figure \ref{fig:ageEffectsRD} point to effects that are large both at younger and older ages among the groups considered, potentially limiting these concerns (at least within the age groups studied).  If we are instead concerned that there may be imbalance over time with later birth cohorts observed until younger ages, we can rule out this concern by considering effects with a balanced sample of birth cohorts, as documented in Table \ref{tab:Gen2TeenBalance}.}  We document this fact in Figure \ref{fig:ageEffects} observing that descriptively, there is a steep gradient in health at birth as mothers age between 15 years until around the mid-20s, at which point the gradient is reversed.  What's more, this pattern is observed to hold even in mother fixed effect models, where identification is drawn off of variation in birth timing within individuals.  A second reason is related to the external validity of these estimates.  In a similar vein to the argument laid out above, if treatment receipt is simply causing individuals to shift births from later in life to earlier in life, and if this is specifically the case for individuals who are negatively selected on health, the estimates we report in this paper may be entirely reversed as women age.  We focus here on these questions, most specifically those of internal validity where we observe relevant information, though comment briefly on questions of external validity.

On the question of internal validity, first we can ask if we actually \textit{do} observe patterns consistent with births being shifted from earlier later to earlier ages.  Descriptively, this does not appear to be the case.  In Figure \ref{fig:fertDesc}, we see that rates of birth generally open up from a young age in treated groups, and then do not show signs of later closing back up.\footnote{These patterns are clearly different to those observed if we focus on other (arbitrary) cut-offs such as above and below median birth weights (Figure \ref{fig:fertDescMed}).}  More formally, we test whether we observe differences in the age composition of births as a result of treatment receipt in a series of balance tests.  We consider this in Table \ref{tab:fertParentLabour}, which consists of a series of regressions of age indicators on treatment receipt within the bandwidth used in Section \ref{sscn:resultgen2}.  When considering the full sample, (column 1), we do not observe evidence to suggest substantial shifts in the age composition of births above and below the cut-off.  This suggests that the selection into fertility is constant across ages rather than shifting individuals from older to younger ages.

\begin{table}[ht]
\caption{The Effect of Compositional Change on Health at Birth}
\label{tab:composition}
\begin{tabular}{lccccccc}
\toprule
  & Gestation & Birth & Size & Infant & Premature & VLBW & Fetal\\
  & Length    & weight &     & Mortality &         &     & Growth  \\
  & (1) & (2) & (3) & (4) & (5) & (6) & (7)\\ \midrule
  \multicolumn{8}{l}{\textbf{Panel A: Birth Timing Only}}\\
  Compositional Effect & 0.00068 & 1.851 & 0.00332 & -0.00001 & -0.000194 & -0.00001 & 0.0474 \\
  Proportional Effect & -0.121 & -0.851 & -0.978 & -1.194 & -0.130 & -0.130 & -1.071 \\ \\
  \multicolumn{8}{l}{\textbf{Panel B: Birth Timing \& Education}}\\
  Compositional Effect & 0.00536 & 8.386 & 0.0177 & -0.00031 & -0.00078 & -0.00022 & 0.207 \\
  Proportional Effect  & -0.953 & -3.856 & -5.213 & -4.816 & -0.522 & -0.492 & -4.677\\ \\
  \multicolumn{8}{l}{\textbf{Panel C: Birth Timing \& Partnership}}\\
  Compositional Effect & 0.00196 & -1.840 & -0.00124 & 0.00001 & -0.00015 & -0.00001 & -0.0509\\
  Proportional Effect& -0.348 & 0.846 & 0.365 & 0.916 & -0.0990 & -0.0147 & 1.150 \\ \\
  \multicolumn{8}{l}{\textbf{Panel D: All}}\\
  Compositional Effect & 0.00673 & 6.544 & 0.0152 & -0.0002 & -0.00067 & -0.00014 & 0.154 \\
  Proportional Effect & -1.196 & -3.009 & -4.477 & -3.148 & -0.449 & -0.301 & -3.479 \\ \bottomrule
  \multicolumn{8}{p{16.4cm}}{{\footnotesize Notes: Compositional effects are estimated as the expected change in outcome if instead of the composition of births among treated individuals, treated births had followed the composition observed among untreated individuals based on estimated age effects from mother fixed effect models documented in Figures \ref{fig:ageEffects}-\ref{fig:ageEffectsCharac}.  Observations used in each column refer to the optimal bandwidth from Table \ref{tab:Gen2BirthOutcomes}  Proportional effect refers to the \% of the estimated treatment effect from Table \ref{tab:Gen2BirthOutcomes} which the compositional effect reflects.  In Panel A, only age compositional changes are considered based on effects in Figures \ref{fig:ageEffects}, while in Panels B-D age effects are considered based on the composition of separate groups documented in Figure \ref{fig:ageEffectsCharac}.}}
  \end{tabular}
\end{table}

It may be the fact that while we do not observe shifts in the total number of births, the composition of who gives birth when changes in a way which is concerning for internal validity.  In examining balance tests by age across a range of characteristics, we also observe little evidence to suggest that there is a broad change in the composition of births across categories such as whether an individual comes from families in which the mother has high education, or in couples with relatively better matches as proxied by the age difference between partners or whether fathers are observed on the birth certificate.  While we observe relatively little evidence of changes in birth timing in this sample, we can consider the relevance of this as a potential mechanism by determining what any compositional changes would imply in terms of birth outcomes if we expressed these changes in terms of mother fixed effect estimates documented in Figure \ref{fig:ageEffects}.  Specifically, we calculate the difference in mean birth outcomes expected if births among treated individuals actually had the same composition as untreated outcomes.\footnote{For example, in Table \ref{tab:fertParentLabour}, we observe that treated individuals are very slightly more likely to be 17 years old, slightly less likely to be 19 years old and so forth. From Figure \ref{fig:ageEffects}, we know that 17 year olds on average have births which weigh around 3200 grams, while 19 year olds have births that weigh around 3250 grams.  We can thus back out the compositional change by mapping the distribution of ages above the cut-off into the distribution of ages below the cut-off.  This is:
\[
\text{Compositional Effect}=\sum_{a=15}^{25} \big\{\left[\Pr(Age=a|Treat=1)-\Pr(Age=a|Treat=0)\right]\times \text{Health at Age }a\big\},
\]
where Health at age $a$ is estimated from mother FE models.}  We observe that such compositional effects are small.  If we consider only changes in birth timing, Table \ref{tab:composition} suggests that compositional changes account for at most 1.2\% of the documented RDD effect.  However, if we consider changes in composition using mother fixed effect models separately by education and partnership status, we observe that compositional changes can account for between around 0.3 to 4.5\% of the documented RDD effect (Table \ref{tab:composition}, panel D) with the lower range of these effects being on measures where treatment effects were observed to be substantial.  Taken together, the results in this section thus suggest that changes in timing are likely of second order importance compared with changes in selection into giving birth.

We cannot directly assess the relevance of concerns related to external validity.  Indeed, as we emphasise in the introduction, it is important to view our results as indicative of the effect of policy receipt on our sample of individuals aged up to the 25 years. Nevertheless, we can draw some relevant information from the findings documented in this section.  We have shown that the effect of compositional changes within the time-window studied is relatively minor.  We have also shown that stocks of health at birth begin to worsen as mother's age increases beyond 25 years.  Finally, we have noted in the previous section that fertility increased by around 0.048 births as a result of treatment receipt.  Taken together, these facts may suggest that births occurring to untreated women above the age of 25 are unlikely to revert the estimates documented here.  However, this is simply suggestive evidence, and whether such a fertility channel remains valid beyond the age of 25 is an open question.

\subsection{Marginal versus Average Effects}
\label{sscn:marginalVsAverage}
The model laid out in Section \ref{scn:modelMethods} and the analysis of channels of intergenerational transmission raises the question of what can be said about the \textit{direct} intergenerational transmission channel, or the `structural' component laid out in Section \ref{scn:modelMethods}.  We have argued above that our evidence shows that there are marginal births which occur among treated women but not among untreated women, and these marginal births obscure our ability to observe the direct effect of treatment receipt.

We can nevertheless ask what these marginal births may look like, and whether this lines up with the evidence discussed previously.  To do so, we will consider `marginal' births as births which would occur if the mother receives treatment, but not if the mother marginally misses out. If we denote the share of marginal births among treated mothers as a proportion $\mu$, with the remaining portion $(1-\mu)$ of births occurring to always-mothers who would give birth whether they received treatment or not.  If we denote as $\alpha^M$ and $\alpha^A$ the causal effect of treatment receipt on marginal and always-mothers conditional on having children, we can decompose the estimated effect of treatment as:
\begin{equation}
    \label{eqn:RDselection}
    \alpha = \mu\alpha^M+(1-\mu)\alpha^A + \mu \Delta,
\end{equation}
where $\Delta$ represents a selection term.  Specifically, this selection effect refers to the difference in average outcomes between marginal and always-treated mothers when they are untreated.  We provide a formal explanation of \eqref{eqn:RDselection} in Appendix \ref{sapp:selectionMA}, along with a graphical explanation in Figure \ref{fig:RDselection}.  Indeed, this derivation is more generally true for any regression discontinuity estimate or treatment effect with an extensive and intensive margin.

To understand what we gain from this decomposition, we can consider what value of $\Delta$ would rationalise treatment effects if $(\alpha^M,\alpha^A)=0$, which is presumably a lower bound for the direct effect of improved health at birth on next generation's health.  For example, consider our estimated effect of treatment receipt on birth weight of -217 grams.  Also, consider that from Table \ref{tab:fertBirths}, we can estimate the proportion of marginal births as $\frac{0.048}{0.235} = 0.20$ (the `extra' births not observed among untreated women).  If we substitute these values into \eqref{eqn:RDselection}, this suggests that $\Delta=\frac{-217}{0.20}=-1064$ grams.  This is illustrative as it suggests that the difference in birth weights between marginal and always treated individuals is substantial, which is what we would expect if treatment receipt was allowing second generation births with the most delicate health stocks to be live births, and hence appear in data.  It is also interesting in that it is \textit{not} consistent with an explanation of smaller changes in birth weight as a result of differences in birth timing (\textit{i.e.}\ it suggests an extensive rather than intensive margin explanation).   More generally however, we can ask what $\Delta$ would have to be to rationalise our treatment effects if $(\alpha^M,\alpha^A)$ vary.  We consider this below in Figure \ref{fig:tauDelta} in which the value of $\Delta$ is plotted as a colour gradient assuming a range of possible values of $\alpha^{M}$ and $\alpha^{A}$.

\begin{figure}[h!]
  \caption{\textbf{Values of $\Delta$ which rationalise $\alpha$}}
  \label{fig:tauDelta}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{"source/Figure 8/selection_BWT.pdf"}
    \caption{Birth weight}
    \label{fig:tauDeltaBW}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.9\linewidth]{"source/Figure 8/selection_VLBW.pdf"}
    \caption{VLBW}
    \label{fig:tauDeltaVLBW}
    \end{subfigure}
    \floatfoot{Notes: Colours presented in heat maps represent values for $\Delta$ which would rationalise treatment effects of $\alpha=-217.7$ (panel A), and $\alpha=0.045$ (panel B), given specific values of $\alpha^A$ and $\alpha^M$ plotted on the $x$ and $y$ axis respectively.  The point indicated by the white arrow refers to the estimate of the intergenerational returns to prenatal Medicaid eligibility in which no fertility effect of the reform was observed.}
\end{figure}


This figure suggests that results line up with our previous posited mechanism of fertility selection in the second generation.  Even if the true intergenerational impacts of early life treatment receipt are quite large, values of $\Delta$ are credible if they represent the births which would have not survived in the second generation.  For example, if we assume that the effects of this policy are similar to those of Medicaid as estimated by \citet{Eastetal2023}, this would imply that marginal births would have needed to be around 1,500 grams lighter than always births had treatment not been received.  Considering that the mean second generation birth weight is 3,157 grams, this suggests that the births which are saved in the second generation are those which are around 2,000 grams.  Similar intuition can be drawn in considering rates of VLBW.  For the estimates from \citet{Eastetal2023}, around 30\% of marginal births would be expected to themselves be VLBW.  Once again, this lines up with the posited mechanism in which the marginal births which are saved are precisely those which have the worst health stocks.  

\section{Conclusion}
\label{scn:conclusion}
In this paper we document a long shadow to public policies, with the impacts of intensive medical care at birth found to have appreciable impacts as much as a quarter of a century later, and to be transmitted by recipients across generations.  We document that unlike a large literature showing virtuous impacts of policies and positive shocks when passed from mothers to their children, we observe that children of treated mothers have worse measured birth outcomes such as low birth weight.  

In examining policy mechanisms, we find that this unexpected result owes to selection into child birth as a result of early life medical care.  Girls who are born weighing just below 1,500 grams, and who receive intensive early life investments are much \emph{more} likely to go on to have their own birth.  We find that policy receipt in generation 1 makes individuals more able to give birth, but on average give birth to babies with weaker stocks of health at birth.  We document a new stylised fact clearly linking birth weight to future fertility (both within and outside of the bandwidth considered in this study), and also make clear that this relationship is modifiable, as treatment receipt is observed to increase rates of birth by around 20\% around VLBW cut-offs.  In many settings where concerns exist about low or below-replacement rate fertility, this suggests another social return to early life investments.  Indeed both this finding and back-of-the-envelope cost-benefit analysis suggests that while in this setting second generation health at birth is observed to be worse on average as a result of policy receipt (owing to the fertility channel), total policy benefits far exceed costs.

It is interesting to consider why these results are different to those generally found in the extant literature.  One reason may owe to external validity.  During the period under study Chile was a middle-income country, and much of the literature examining intergenerational spillovers is drawn from a high-income setting. What's more, data demands in following individuals across generations are high, and our analysis is limited only to mothers aged up to 25.  Thus, our evidence must be viewed through this lens, as we are unable to observe transmission among older mothers, or from fathers to their children.  However, there is reason to believe that these findings may be relevant to similar groups in broader settings.  Estimated impacts of this specific early life treatment on both first generation health and later life educational outcomes has been shown to be comparable in Chile to Norway, Denmark, and the US \citep{Bharadwajetal2013,Daysaletal2022,Almondetal2010}, suggesting some external validity in other outcomes in this domain.  The relevance of an extensive (fertility) and intensive margin (health at birth) policy response seems to be key in explaining differences with existing results.  The only other study examining intergenerational spillovers of health policies of which we are aware \citep{Eastetal2023} is focused on a specific low-income group of recipients and finds no fertility response, and subsequent positive intergenerational spillovers.  A similar lack of fertility response has been documented in \citet{Nilsson2017} when considering alcohol consumption, suggesting that the degree to which extensive and intensive margin policy effects come into play and why is a relevant question to consider.

These findings have implications related to the ways downstream public health and welfare policies are defined following initial policy receipt.  While a number of influential papers show that the impacts of early life treatments are unambiguously positive for their recipients \citep{Bharadwajetal2013,Almondetal2010,Eastetal2023}, suggesting the need for compensatory policies for individuals who marginally miss out on such policies, our results also point to the importance of reinforcing investments, at least when considering the second generation of the original policy recipients in this sample.




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  {\Large
  \textbf{Online Appendices for:}} \\ \vspace{2mm}
  {\large\textbf{``Estimating Intergenerational Returns to Medical Care: New Evidence from At-Risk Newborns''}} \\  \vspace{2mm}
 {\large Damian Clarke \qquad Nicol\'as Lillo Bustos \qquad Kathya Tapia-Schythe}
 
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\section{Data Appendix}
\label{app:data}
In this appendix we collect additional information related to data and descriptive statistics referred to in the paper.  Sub-section \ref{sscn:limits} notes further information related to data matching in administrative datasets, and \ref{sscn:descriptives} collects all descriptive figures in the order in which they are referred to in the text.

\subsection{Matching Microdata}
\label{sscn:limits}
While individual microdata files are high quality and comprehensive, matching between microdata files requires access to restricted information---namely each individual's RUT, or national identity number.  Fortunately, for a number of years Chile's Ministry of Health provided access to a unique anonymised version of this RUT which were harmonised across administrative records held by the Ministry of Health.  This allows for us to match all administrative records in this paper (namely births, deaths, and inpatient hospitalisations) across files such that we generate a complete history for each individual.  We can do this for the period of 1992-2018. This anonymised version of the RUT is generated by the Ministry of Health to match across \textit{their own} administrative records, and we use this to generate the full data used in this paper.  Nevertheless, the use of these matched microdata files implies two specific limitations which we lay out below.

\paragraph{Limitation 1: Matching Data with Administrative Data Held by Other Ministries}  The anonymised version of the RUT does not allow us to match health records with data held by other government authorities.  Thus, we cannot observe information on later educational attainment for all births (which requires links to administrative records held by the Ministry of Education), nor their eventual partnership choices (which requires links to administrative records held by the Civil Registry).  Such intra-agency data collaborations require formal collaboration agreements, and resultingly matched microdata cannot be shared.  One benefit of using the anonymised version of the RUT generated by the Ministry of Health is that all matched data can be publicly available given that (i) no sensitive information is revealed in this process and (ii) all data and variables were originally shared publicly on their open data page (\url{https://deis.minsal.cl/#datosabiertos}) and are stored on internet archives such as the Wayback Machine (see, e.g.\ \url{https://web.archive.org/web/20190326212649/https://deis.minsal.cl/#datosabiertos}).

\paragraph{Limitation 2: Temporal Access 1992--2018} A second limitation relates to the period which we can study.  In approximately 2020, the Ministry of Health changed their data sharing protocols and no longer publish these administrative databases with anonymised versions of the RUT.  Because these microdata files are shared with around 1 year's delay, publicly shared data from 2019 onwards does not include an anonymised version of the RUT, meaning that we cannot match mothers from 2019 onwards with their own birth weights when they were born.  Indeed, more generally the data sharing standards of the Ministry became more restrictive, and birth weight itself is no longer reported exactly, but rather reported at the individual level in categorical groups.  Thus, we are limited with our analysis to consider periods from 1992--2018, implying that we can follow women who have turned a (maximum) of 26 years. Given that we observe no individuals who have completed their 26\textsuperscript{th} of life, we limit our sample to mothers aged up to 25.  This results in the removal of a small number of individuals who have turned 26 though not yet reached the end of their 26\textsuperscript{th} year, though our results are not sensitive to this choice.



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\subsection{Descriptive Figures and Tables}
\label{sscn:descriptives}
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\begin{figure}[h!]
\caption{\textbf{Schematic Representation of a Lineage}}
\label{fig:lineage}
\begin{tcolorbox}[title=A lineage]
\begin{center}
\textbf{``Grandmother''} \hspace{1.7cm} \textbf{``Mother''}  \hspace{1.7cm} \textbf{``Children''} \\
$N=3,010,251$ \hspace{1cm} $N=3,010,251$  \hspace{1.1cm} $N=420,389$ \\
\hspace{9mm} \Tshirt{5} \hspace{9mm}\raisebox{2\baselineskip}{\tikz[>=stealth]\draw[very thick,black,->](0,0)--(.05\textwidth,0);}
\hspace{9mm}\Tshirt{5}\hspace{9mm}\raisebox{2\baselineskip}{\tikz[>=stealth]\draw[very thick,black,->](0,0)--(.05\textwidth,0);} \hspace{9mm}\Tshirt{4} \\
\hspace{5cm} \emph{1\textsuperscript{st} Generation birth} \hspace{6mm} \emph{2\textsuperscript{nd} Generation birth} 
\end{center}
\end{tcolorbox}
\floatfoot{Notes:  Birth certificates capture characteristics of the individual born, as well as their parents, and we observe the unique identity number of the parent as well as the individual born. Thus, for ``Mothers'', at their time of birth we observe their birth outcomes, as well as their mothers' characteristics (``Grandmothers'').  While for ``Children'', we observe their birth outcomes, their mother's birth outcomes, and their mother's outcomes at the time the child was born.  The nomenclature ``mother'' refers to all girls who could potentially go on to being mothers, many of which have not yet given birth. Observations refer to quantities used in final estimation sample of mother's aged 15-45 at birth and without missing observations in covariates}
\end{figure}




  \begin{table}[htpb!]
    \centering
    \caption{\textbf{Matched Observations between Microdata Registers}}
    \label{tab:matches}
    \begin{tabular}{lcccc} \toprule
      Register & Observations with & Matched to & Matched to  & Matched to \\
               & Valid Unique IDs  & Births     & Hospitalisation & Deaths \\ \midrule
      1992--2018  Births          & 6,617,637  & 435,013    & 5,656,409 & 83,841 \\
      2001--2019  hospitalisation & 27,995,452 & 2,924,795 & ---       & 1,272,340 \\
      1992--2018  Deaths          & 2,393,583  & 83,841    & 4,627,999 & ---    \\ \midrule
      \multicolumn{5}{p{14.8cm}}{\footnotesize Notes: Column 1 presents the
        total number of valid observations in each dataset. Column 2 notes the number of births which match to each dataset.  In the case of the birth register, it refers to the number of births which match to other births in the data (\textit{i.e.}\ mother--child links). Column 3 notes the number of hospitalisations which match to each other database.  Note that in the case of births, the number of hospitalisations linked to births is not the same as the number of births linked to hospitalisations in the preceding column given that a single birth can be hospitalised multiple times.  Finally, column 4 notes the total number of deaths which are matched with births occurring in the sample.}
    \end{tabular}
  \end{table}

 \begin{landscape}
  \begin{table}[tpb!]
    \centering
    \caption{\textbf{{Temporal Links between Mother--Child Matched Birth Years}}}
    \label{tab:birthChart}
    \begin{tabular}{lccccccccccccccc} \toprule
    Mother & \multicolumn{15}{c}{Child} \\ \cmidrule(r){2-16}
    Year & 2004 & 2005 & 2006 & 2007 & 2008 & 2009 & 2010 & 2011 & 2012 & 2013 & 2014 & 2015 & 2016 & 2017 & 2018 \\ \midrule
1992&10&57&346&1,475&3,789&6,205&7,965&9,468&9,964&9,898&9,831&9,438&8,954&8,663&8,912\\
1993&0&6&60&399&1,622&3,806&5,869&7,632&8,962&9,344&9,826&9,413&8,578&8,162&8,260\\
1994&0&0&7&60&373&1,614&3,608&5,982&7,429&8,457&9,273&9,023&8,519&7,974&7,865\\
1995&0&0&1&11&76&418&1,437&3,498&5,560&6,654&8,000&8,298&7,770&7,480&7,235\\
1996&0&0&0&3&9&83&356&1,484&3,493&4,967&6,205&7,046&7,305&7,068&6,774\\
1997&0&0&0&0&0&10&61&400&1,442&3,183&4,684&5,433&5,687&6,123&6,478\\
1998&0&0&0&0&0&0&5&56&367&1,431&2,911&3,882&4,366&4,899&5,566\\
1999&0&0&0&0&0&0&2&7&52&369&1,331&2,349&2,914&3,415&4,081\\
2000&0&0&0&0&0&0&0&2&6&56&326&1,057&1,879&2,270&2,736\\
2001&0&0&0&0&0&0&0&0&0&3&49&261&754&1,297&1,638\\
2002&0&0&0&0&0&0&0&0&0&0&4&43&248&582&1,022\\
2003&0&0&0&0&0&0&0&0&0&0&1&7&23&166&471\\
2004&0&0&0&0&0&0&0&0&0&0&0&1&6&48&177\\
2005&0&0&0&0&0&0&0&0&0&0&0&0&1&4&28\\
2006&0&0&0&0&0&0&0&0&0&0&0&0&0&0&3\\
    \bottomrule
    \multicolumn{16}{p{20.3cm}}{{\footnotesize Notes: Values note the total number of mothers who are born in each of the years indicated in the left-hand column giving birth in each year indicated by row headers. All values are based on matched administrative records, and are provided from 1992 (the first birth year observed) up to 2006 (the last birth cohort in which a later-born child was linked to a mother born in this year.) A small number of observations ({11}) with inconsistencies in mother's age are removed prior to estimation.}}
    \end{tabular}
  \end{table}
\end{landscape}
  


\begin{table}[htpb!]
  \caption{\textbf{Summary Statistics -- Births local to the 1,500 gram threshold}}
  \label{tab:sumstatsClose}
  \begin{tabular}{lccccc} \toprule
    & Obs.\ & Mean & Std.\ Dev.\ & Min.\  & Max. \\ \midrule
    \multicolumn{1}{l}{\textbf{{Panel A: First Generation Births}}} &&&&& \\
    \input{tables/Appendix/SummaryB_G1}
    &&&&&\\
    \multicolumn{1}{l}{{\textbf{Panel B: Second Generation Births}}} &&&&& \\ 
    \input{tables/Appendix/SummaryB_G2}
    \bottomrule
    \multicolumn{6}{p{15.4cm}}{\footnotesize Notes: Summary statistics are displayed for births occurring close to the 1,500 gram treatment threshold for the first generation (all births between 1992 and 2018), as well as those births matched to prior births (second generation births).  The full sample consists of all births occurring between 1992 and 2018 in Chile from administrative data maintained by the Ministry of Health, and here we subset based on treatment assignment (mother's birth weight).  This is based on the 134.4 and {227.5} gram optimal bandwidth cut-off when considering infant mortality for the first generation and Anderson index for the second generation.  Hospitalisation days refer to days \emph{only} for those births which are admitted to hospital.}
  \end{tabular}
\end{table}

\begin{figure}[htbp!]
    \caption{\textbf{{Age of Mothers at Childbirth}}}
    \label{fig:Agemothers}
      \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/nacimientos_edadmadre1992.eps}
      \caption{1992}
      \label{fig:Agemothers_a}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
        \includegraphics[width=1\linewidth]{figures/Appendix/nacimientos_edadmadre2018.eps}
      \caption{2018}
      \label{fig:Agemothers_b}
    \end{subfigure}

    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/nacimientos_edadmadre.eps}
      \caption{1992-2018}
      \label{fig:Agemothers_c}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
        \includegraphics[width=1\linewidth]{figures/Appendix/nacimientos_edadmadreG1.eps}
      \caption{Generation 1 Mothers}
      \label{fig:Agemothers_d}
    \end{subfigure}
      \vspace{-6mm}
      \floatfoot{Notes: The distributions of births by mother's age are displayed.  Births with mothers greater than age 25 are shown in green, while mothers aged 25 and below (corresponding to the maximum age in the intergenerational sample) are shown in red. }
  \end{figure}

  
   \begin{figure}[htpb!]
    \caption{\textbf{Observed Birth Spacing Between Children and their Future Siblings}}
    \label{fig:birthSpace}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/birthSpacingAll.eps}
      \caption{All Individuals}
      \label{fig:bSpaceAll}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/birthSpacingClose.eps}
      \caption{$1300\leq$ Birth weight $\leq 1700$}
      \label{fig:bSpaceClose}
    \end{subfigure}
    \floatfoot{Notes: Histograms document the space in years between each observed birth and following births of each mother  (provided following births are observed) in the full sample of administrative records from birth registries (panel (a)), and only those births with babies weighing between 1,300 and 1,700 grams (panel (b)).}  
    \end{figure}
\clearpage

\end{spacing}


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\section{Identification Checks}
\label{app:RDvalidity}
In this appendix we collect a range of tests relating to the validity of the very low birth weight RDD.  Identification in a regression discontinuity design relies on the well-known continuity assumption related to unobservable factors around the cut-point \citep{Hahnetal2001}.  If all unobservable factors are balanced on either side of the discontinuity, provided a small enough bandwidth is used for estimation, and provided that the cut-off generates a discontinuous jump in treatment, RD models can be used to isolate the impact of treatment on the dependent variable(s) of interest.   In this appendix we first collect tests which document the relevance of the treatment in the first generation (Appendix \ref{scn:resultsGen1}), and then a series of tests examining the identifying assumptions relating to treatment assignment to generation 1 mothers.



\subsection{Proof of Treatment}
\label{scn:resultsGen1}
Prior to considering any second generation outcomes, we seek to confirm previous evidence of the relevance of this treatment in this setting. This is a `first-stage' which establishes the feasibility of later observing \emph{intergenerational} spillovers of the initial treatment.  \citet{Bharadwajetal2013} have demonstrated that assignment to intensive neonatal care regimes due to crossing the VLBW threshold brought about sharp reductions in rates of infant mortality in Chile.  We first document that these results can be replicated with the newly public matchable microdata files, on various samples of data, and using both their original and more recent optimal RD methods.  We also document that these results are found among earlier generations, which is key in demonstrating the relevance of these treatments among cohorts who will be mothers in the intergenerational sample.  
  
Figure \ref{fig:IMR} presents regression discontinuity plots which examine binned rates of infant mortality among births occurring in Chile around the 1,500 gram cut-off.  Panel (a) replicate \citet{Bharadwajetal2013}'s methods and definitions, working with the same sample of births between 1992-2007.  Note that here \citet{Bharadwajetal2013} bin weights in blocks of 30 grams, centred on points of 10 grams, with the exception of the point closest to the cut-off on either side, which is defined sharply.  This implies that a single birth will be represented in multiple points.  We follow their methods for comparability in panel (a), considering the same 100 gram range on either side of the cut-off, and in panels (b) and (c) use 10 mutually exclusive bins on either side of the treatment cut-off, and optimal bandwidth choices of \citet{Cattaneoetal2020}.  In all cases, a clear increase in infant mortality is observed when crossing the 1,500 gram threshold.    

For our purposes, what is key is ensuring the relevance of initial treatment, which is clear even graphically in Figure \ref{fig:IMR}.  Formal RD estimates are presented in Appendix Table \ref{tab:IMRgen1}. Estimates are presented based on all three samples presented in Figure \ref{fig:IMR},  estimating that, in the `intergenerational sample', infant mortality falls by 2.7 pp, compared to a base of 13.8pp, or by approximately 22\%, clearly establishing the relevance of these treatments in this particular setting. If we alternatively examine only earlier years, for example 1992--2001, similar results are observed.  See Figure \ref{fig:IMR_altyears}.  If extending these tests to consider rates of infant death up to 2018, the results first documented in \citet{Bharadwajetal2013} are, if anything, strengthened, with column 3 of Table \ref{tab:IMRgen1} estimating a 2.7 pp reduction in rates of infant death against a base of 10.2 pp, even within the very tight bandwidth of 100 grams used by \citet{Bharadwajetal2013}, with very similar results observed in column 4 when estimating based on the optimal bandwidth of 134.4 grams. Note that these effect sizes are large, of the order of magnitude of a 20-30\% decline in rates of infant death. A simple back of the envelope cost-benefit calculation presented in Online Appendix \ref{sscn:discussion} suggests that these benefits from treatment receipt in the first generation dwarf any future costs incurred as a result of the policy given required investments in future generations.  This holds even if we additionally allow for negative spillovers from fathers to their children.


\paragraph{Estimation Sample}
As we noted in section \ref{scn:background}, and as laid out in \citet{Bharadwajetal2013}, there is clear evidence of a sharp cut-off in treatment rules at 1,500 grams when babies are born at 32 weeks or above.  In certain cases, babies born at below 32 weeks will access the same treatment regimes, whether they are born at above or below the 1,500 gram threshold.  In main estimates we thus focus exclusively on the sample of individuals born at 32 weeks and above, and in which case the discontinuity certainly binds. This sample covers 56\% of all births within 200 grams of the 1,500 gram threshold.  However, there is some evidence to suggest that at least in a more limited way, there is a discontinuity in treatment around 1,500 grams for babies born at below 32 weeks.  We document this in Figure \ref{fig:RDhealthU32} which shows rates of hospitalisation by age, with evidence that---in line with policy definitions---babies born at less than 1,500 grams have elevated rates of hospitalisation early in life, and in particular in the first year of life, consistent with more intensive treatment. Thus, as a robustness check, we additionally present results from the main text covering the full sample of individuals, regardless of their gestational length.  


\subsection{Identification Checks}\label{sscn:IDtests}
Identifying the impacts of marginal medical investments at birth on subsequent outcomes based on our RD design requires that no other factors vary sharply local to the 1,500 gram cut-off.  We consider a number of tests of these identifying assumptions, though note that this design has previously been validated in this and other settings by \citetappendix{Bharadwajetal2013,Chynetal2021,Almondetal2010,Daysaletal2022}.  In particular, our tests here focus on two considerations: firstly, is there balance at treatment receipt of observable characteristics, and secondly, is there evidence of manipulation of birth weights suggesting that there may be systematic, or strategic, selection into treatment by medical practitioners, or by families of babies born in this bandwidth.

A first consideration which is key is examining the predetermined characteristics of individuals who are located just below the 1,500 gram threshold, and hence receive of intensive medical intervention, and those located just above the threshold.  We would be concerned if we observed that certain individuals are more likely to receive access to the policy, as it may illustrate imperfect compliance with the treatment threshold, and confound estimates presented throughout this paper.  We examine a number of measures of first generation mothers, fathers and births, presented graphically in Appendix Figure \ref{fig:Balance}, with associated RD tests in Appendix Table \ref{tab:balance}.  Specifically, we do not see that treated individuals have parents of substantially different ages or educational levels or who are more closely aligned in age, nor do we see that they are more likely to be married, or are more likely to be born in urban areas.  This balance point is also highlighted by \citetappendix{Bharadwajetal2013}. 

Nevertheless, a general concern in this case is still related to manipulation of the running variable, potentially not captured by these observable factors.  If parents or medical practitioners were able to manipulate the running variable (officially recorded birth weight), they may be able to selectively ensure coverage for certain types of individuals.  If this is the case, and if such manipulation owed to visual clues or information from patient histories inferred by medical practitioners, one may suspect that babies registered as having weights just to the left of the cut-off look different in birth size or gestational length than those just to the right.  In Figure \ref{fig:observableManip} we observe no evidence to suggest that this is the case. We also note that in the case of manipulation, one would expect a greater likelihood of observing births at points close below 1,500 grams, and a lower likelihood of observing births just above 1,500 grams, which is not something we observe. As proposed by \citetappendix{Almondetal2011}, we conduct tests formally examining whether there are more births observed in micro-data registered as having birth weight just to the left compared to just to the right of the cut-off, and find no evidence to suggest that this is the case (Appendix Table \ref{tab:Almondtest}).

Another specific concern related to manipulation and measurement in this setting, discussed extensively in the existing literature, is the presence of heaping in birth weight \citepappendix{Barrecaetal2011,Barrecaetal2016,Almondetal2011}. This can be observed when examining simple plots of the frequency of individuals observed at particular birth weights.  In Appendix Figure \ref{fig:BWdesc} we observe that while birth weight is regularly distributed when zooming out across the entire distribution, regular peaks are observed at 50 and 100 gram bins, apparent when zooming in on birth weight in panel (b).  Similarly, \citet{Bharadwajetal2013} note that this rounding occurs differentially in certain types of hospitals, implying that it may be not be innocuous in estimation.  We note that in the case of assignment rules, individuals will receive treatment only if they are observed to have a birth weight just \emph{below} 1,500 grams. Given this, throughout this paper we control flexibly for heaping and include controls for demographic factors proposed by \citetappendix{Bharadwajetal2013}.  But to ensure that our estimates are not driven by heaping, specifically that at 1,500 grams, we estimate the `Donut' RD models suggested by  \citetappendix{Barrecaetal2011,Barrecaetal2016}, removing observations which are located very close to the cut-off.  We discuss these models when discussing robustness of our estimates in Section \ref{sscn:robustness}.

\clearpage

\begin{figure}[htpb!]
    \caption{\textbf{Power Analysis -- RD Models of Intergenerational Impacts}}
    \label{fig:PowerGen2}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_SEMANAS_o32_Power.eps}
      \caption{Child's Gestational Length, Baseline}
      \label{fig:PWeeksB}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_SEMANAS_o32_heaping_Power.eps}
      \caption{Child's Gestational Length, Heaping Control}
      \label{fig:PWeeksH}
    \end{subfigure}
    
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_PESO_o32_Power.eps}
      \caption{Child's Birth weight, Baseline}
      \label{fig:PWeightB}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_PESO_o32_heaping_Power.eps}
      \caption{Child's Birth weight, Heaping Control}
      \label{fig:PWeightH}
    \end{subfigure}
    
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_TALLA_o32_Power.eps}
      \caption{Child's Birth length, Baseline}
      \label{fig:PLengthB}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/NAC_TALLA_o32_heaping_Power.eps}
      \caption{Child's Birth length, Heaping Control}
      \label{fig:PLengthH}
    \end{subfigure}
    \floatfoot{Notes: Curves in each figure plot the power of our regression discontinuity design to detect effect sizes indicated by Tau on the horizontal axes against a null of zero effects.  Effects refer to intergenerational impacts, and as such, in each case, samples consist of second generation births corresponding to our principal estimation sample.  Power calculations follow \citet{Cattaneoetal2019b}.}
  \end{figure}


\begin{landscape}
\begin{figure}[htpb!]
  \caption{\textbf{Birth weight Assignment Thresholds and Infant Mortality}} 
  \label{fig:IMR}
  \begin{subfigure}{.33\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_BLN_19922007_despues.eps}
    \caption{BLN Method and Sample}
    \label{fig:IMRBLN32}
  \end{subfigure}
  \begin{subfigure}{.33\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922018.eps}
    \caption{Full Period and Optimal Bandwidth}
    \label{fig:IMROPT32}
  \end{subfigure}
  \begin{subfigure}{.33\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922001.eps}
    \caption{Early Cohorts and Optimal Bandwidth}
    \label{fig:IMROPT32}
  \end{subfigure}
  \floatfoot{Notes: Each sub-plot examines the impact of crossing the VLBW threshold on infant mortality.  All panels present estimates for gestational weeks 32 and above (where assignment rules clearly apply). Panel (a) replicates \citet{Bharadwajetal2013}'s methods using overlapping (30 g) bins and a 100 gram bandwidth.  Panel (b) uses optimal bandwidth selection methods of \citep{Calonicoetal2020a}, and plots a quadratic fit with 95\% CIs, where point sizes are indicative of relative sample sizes. Panel (c) replicates the optimal plot from panel (b) but focusing only on earlier birth cohorts, who are represented as mothers in the intergenerational sample.}
  \end{figure}
\clearpage      
\end{landscape}

\begin{landscape}
\begin{table}[htpb!]
    \centering
    \caption{\textbf{Initial Policy Impacts on Infant Mortality Rates}}
    \label{tab:IMRgen1}
    \begin{tabular}{lcccccc}
    \toprule
    & \multicolumn{2}{c}{\citet{Bharadwajetal2013}} & \multicolumn{2}{c}{Full Period}
    & \multicolumn{2}{c}{Early cohorts} \\
    & \multicolumn{2}{c}{Period} &\multicolumn{2}{c}{(1992-2018)}&\multicolumn{2}{c}{(1992-2001)} \\
    \cmidrule(r){2-3} \cmidrule(r){4-5} \cmidrule(r){6-7}
    & (1) & (2) & (3) & (4) & (5) & (6)\\ \midrule
   \input{tables/Appendix/tableIMR}
    \\ \midrule
    \citet{Bharadwajetal2013} Procedure & Y &  &Y & & Y & \\
    Optimal Bandwidth RBC & & Y &  &Y & & Y \\ \bottomrule
    \multicolumn{7}{p{18.2cm}}{{\footnotesize Notes: Each column presents estimates of the impact of intensive treatment owing to the VLBW assignment on an indicator for infant mortality.  Models in odd columns replicate the methods of \citet{Bharadwajetal2013} using a 100 gram cut-off, and in even columns use the MSE optimal bandwidth of \citet{Calonicoetal2020a}.  In each case standard RD models are presented (top rows) which correspond to methods implemented by \citet{Bharadwajetal2013}, and robust bias corrected estimates and standard errors are presented below.  Columns (1) and (2) use the exact same time period as that studied by \citet{Bharadwajetal2013} (1992-2007), columns (3) and (4) consider the full sample of data used in this paper, and columns (5) and (6) consider only birth cohorts in which second generation mothers are born. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}
\end{table}
\end{landscape}

\begin{figure}[htpb!]
  \caption{\textbf{Birth weight Assignment Thresholds and Infant Mortality (Early Years Only)}} 
  \label{fig:IMR_altyears}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922001.eps}
    \caption{$\leq$ 2001 and Optimal Bandwidth}
    \label{fig:IMRBLN32}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922004.eps}
    \caption{$\leq$ 2004 and Optimal Bandwidth}
    \label{fig:IMROPT32}
  \end{subfigure}
  
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922005.eps}
    \caption{$\leq$ 2005 and Optimal Bandwidth}
    \label{fig:IMROPT32}
  \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/imrt_o32_optimal_19922006.eps}
    \caption{$\leq$ 2006 and Optimal Bandwidth}
    \label{fig:IMROPT32}
  \end{subfigure}
\floatfoot{Notes: Refer to Notes to Figure \ref{fig:IMR}.  Identical plots are presented to panel (c), however here documenting alternative considerations of ``early cohorts'', as indicated in captions.}  
  \end{figure}
\clearpage      

\begin{figure}[h!]
  \caption{\textbf{Discontinuities in Hospitalisation in Babies Born at $<$ 32 Gestational Weeks}}
  \label{fig:RDhealthU32}
    \centering
    \includegraphics[width=0.7\linewidth]{figures/Appendix/hospitalization_u32.eps}
    \vspace{-8mm}
    \floatfoot{Notes: Each point estimate and confidence interval refers to the impacts of early life medical investment on an individual's days of hospitalisation at each age, estimated by RD using the full sample of births \emph{born at less than 32 gestational weeks} using the RD specification.  Thicker black error bars present 90\% CIs, while thinner error bars report 95\% CIs.  Days of hospitalisation are measured as totals for all individuals who have reached the age indicated, and take the value of 0 if the individual is not hospitalised this year, or otherwise a positive integer reporting the total number of days spent in hospital.  All estimates follow the procedures laid out in section \ref{scn:modelMethods}, and report RBC estimates using a local linear regression with a triangular kernel in the MSE optimal bandwidth.} 
\end{figure}




\begin{landscape}
  \begin{figure}[htbp!]
    \caption{{\textbf{Balance Tests -- Generation 1 Family and Birth Characteristics}}}
    \label{fig:Balance}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_ageMum.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_educMum.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_employedMum.eps}
    \end{subfigure}   

    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_ageDad.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_educDad.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_employedDad.eps}
    \end{subfigure}   
    

    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_marrieda.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_difedad.eps}
    \end{subfigure}
    \begin{subfigure}{.32\textwidth}
      \centering
      \includegraphics[width=0.88\linewidth]{figures/Appendix/figureA3_RDplot_urbano.eps}
    \end{subfigure}
    \floatfoot{Notes: Plots document balance tests examining characteristics of individuals within the \citet{Calonicoetal2020a} RBC optimal bandwidth of the 1,500 gram treatment threshold. Binned averages for each outcome are presented in 20 gram birth weight bins, with circle sizes documenting the relative proportion of individuals in each bin.  A separate split quadratic and confidence intervals are documented on each side of the cut-off.  Formal tests of discontinuities are presented in Table \ref{tab:balance}.}    
    \end{figure}
\end{landscape}

\begin{landscape}
\begin{table}[htpb!]
  \caption{{\textbf{Balance Tests -- Generation 1 Family and Birth Characteristics}}}
  \label{tab:balance}
  \scalebox{0.95}{
  \begin{tabular}{lcccccccccc} \toprule
    &\multicolumn{3}{c}{Mother} & \multicolumn{3}{c}{Father} & & {Parent's} & {Observed} & Urban\\ \cmidrule(r){2-4}\cmidrule(r){5-7}
    & Age & Education &  {Employed} & Age & Education &  {Employed} & Married &{Age Diff.}\  & Dad & Status\\ \midrule
    \input{tables/Appendix/tablefigureA3}
    \bottomrule
    \multicolumn{11}{p{23.2cm}}{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for newborns.  Each outcome refers to mother, father, or family level characteristics of all first generation observations, and as such are balance tests examining assignment to treatment. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}. Robust bias corrected standard errors clustered at the gram level are reported in parentheses. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}
    \end{tabular}}
\end{table}
\end{landscape}



\begin{figure}[htpb!]
    \caption{\textbf{Observable Birth Outcomes by Birth Weight}}
    \label{fig:observableManip}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/sizeByWeight.eps}
      \caption{Birth size (cms)}
      \label{fig:observableManipSize}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
        \includegraphics[width=1\linewidth]{figures/Appendix/gestByWeight.eps}
      \caption{Gestational weeks}
      \label{fig:observableManipWeeks}
    \end{subfigure}
    \floatfoot{Notes: Scatter plots are presented where each point represents average birth size (panel (a)), or gestational weeks (panel (b)) in 50 gram birth weight bins.  Each panel is based on the full sample of births from 1992--2018 between 1,300-1,700 grams, and point sizes reflect the relative frequency of the sample in each bin.  Separate linear trends are plotted on each side of the 1,500 gram VLBW threshold.}
\end{figure}

\begin{figure}[htpb!]
    \caption{\textbf{Birth Weight Frequency in Administrative Records}}
    \label{fig:BWdesc}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/histFull2007.eps}
      \caption{Full Sample}
      \label{fig:BWdescfull}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
      \centering
      \includegraphics[width=1\linewidth]{figures/Appendix/histShort2007.eps}
      \caption{1300-1700 grams}
      \label{fig:BWdescshort}
    \end{subfigure}
    \floatfoot{Notes: Density plots are presented based on the full sample of births from 1992--2018.  Panel (a) includes all births, while panel (b) limits only to births recorded as weighing between 1300 and 1700 grams (inclusive).  In panel (b) 10 gram bins are plotted in the histogram.}
\end{figure}


\begin{landscape}
\begin{table}[hptb!]
\centering
\caption{\textbf{Testing for birth heaping at the cut-off}}
\label{tab:Almondtest}
\scalebox{0.95}{
\begin{tabular}{lcccccccccc}\toprule
& (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) \\ \midrule
\multicolumn{11}{l}{\textbf{Panel A: Total Difference}}\\
\input{tables/Appendix/AlmondTest} \midrule
\multicolumn{11}{l}{\textbf{Panel B: Total Difference and Distance Trend}} \\
\input{tables/Appendix/AlmondTest_trend}
\bottomrule
\multicolumn{11}{p{24.8cm}}{{\footnotesize Notes: Each column presents a regression of the number of births observed on an indicator for whether the recorded weight is below 1,500 grams, in a fixed range around the 1,500 gram cut-off.  This range is varied across columns from$\pm$ 20 grams to $\pm$ 400 grams, as indicated in the table footer.  In panel (A) a single binary ``$<$ 1,500 gram'' indicator is included, while in panel (B) this is additionally interacted with a control for absolute difference in grams, to control for the fact that the number of births gradually increase as birth weight increases. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
\end{tabular}}
\end{table}
\end{landscape}






\clearpage


  \clearpage
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\section{Additional Model Details}
\label{app:model}
\subsection{A Single Generation Model}
\label{sapp:singleGenModel}
Previous literature in this setting \citepappendix{Bharadwajetal2013,Chynetal2021} has proposed a first generation model consisting of initial endowments, medical treatments (which interact with arbitrary treatment assignment cut-offs), and subsequent post-hospital investments. 
To understand the impacts of treatment receipt on a treated individual during their own life, consider the stylised setting where the outcome of interest is an individual's (own) health.  In a simple framework, along the lines of \citetappendix{Grossman2000}, suppose that these time-varying health measures denoted $H_{it}$ depend on initial endowments ($H_i$), medical intervention at birth ($D_i$), as well as subsequent parental investment.\footnote{In Grossman's canonical health capital model, health at time $t+1$ depends upon health at time $t$ as well as (recursively) health at birth.  Here given we are interested in estimating impacts of early life interventions we focus principally on health at birth.  However, as we show below, we can incorporate dynamic health flows in this model without greatly altering the implications of this model for understanding RDD estimates.}   As \citetappendix{Bharadwajetal2013} note, parental investment can interact with initial medical treatments in such a way to reinforce or compensate initial treatment receipt.  Following their notation, parental investment is denoted $I_t^{post}(H,D,\zeta)$, capturing accumulated investments up to $t$ which may be a function of initial health, treatment at birth, and subsequent shocks $\zeta$.  Thus, `first generation' health is modelled as:
\[
    H_{it}=\phi_tH_i + \psi_tD_i + \varphi_tI_t^{post}(H_i,D_i,\zeta_i)+ X_{it}\beta_t + \upsilon_{it},
\]
where it is clear that individual health at time $t$, $H_{it}$ may depend on initial treatment directly, but also in the way that this treatment $D_i$ interacts with subsequent parental behaviour.

As in \citetappendix{Bharadwajetal2013}, while the RD isolates an exogenous shift in treatment intensity, the coefficient of interest from \eqref{eqn:baseline} when considering $H_{it}$ as the outcome of interest captures the full policy impact up until $t$:
\[
    \widehat\alpha = \psi_t\cdot\kappa + \varphi_t\cdot \Delta I_t^{post}(c),
\]
consisting of both structural effects from treatment at birth, as well as differences in posterior parental investments surrounding the treatment cut-off.

Thus, when considering health outcomes of treated women after birth but prior to any second generation births, the estimated coefficient $\widehat\alpha$ will be interpreted as the total policy relevant treatment effect, based on individual and parental response up until the moment that the outcome of interest is measured.  

\subsection{Greater Granularity in Fertility Choices}
\label{sapp:ferttime}
In Section \ref{sscn:model} we develop a model in which selection is allowed to differ when considering fertility at younger and at older ages.  This definition can be further generalised without any major impacts on the modelling implications insofar as they drive our consideration of potential mechanisms.  To see this, note that instead of \eqref{eqn:selec1a}-\eqref{eqn:selec1b} we could write age-specific fertility as a function of individual's health, early life investments, parental investments and other behaviours as:
\begin{eqnarray}
  \label{eqn:selecAge}
  \text{Fertility at Age X}^{*}_i &=& \gamma_1^{X} H_i + \gamma_2^{X} D_i + \gamma_3^X I^{post}(H_i,D_i) + \gamma_4^X B^{post}(H_i,D_i) + \upsilon^X.
\end{eqnarray}
for all $X$ observed.  A key point here is that this permits each of these determinants to affect fertility in a different way throughout an individual's life.  In this case, if birth weight is modelled as in \eqref{eqn:selec2}, this will lead to a separate expected health outcome at birth for each age $X$, written as:
\begin{eqnarray}
E[BW_{ij}|\text{Fertility at Age X}_i=1]&=& H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(\cdot) + \varPsi B^{post}(\cdot) \nonumber \\ &&+  
\rho_{X}\sigma\lambda_{X}\left[H_i,D_i,I^{post}(\cdot),B^{post}(\cdot)\right], \nonumber
\end{eqnarray}
and now, selection into fertility may vary \textit{at each age} $X$ via distinct selection terms $\lambda_X(\cdot)$.  The implications of this would be identical to those discussed at the end of Section \ref{sscn:model}, simply with a richer channel to describe birth timing.

\subsection{Dynamic Health Stocks}
\label{app:dynamicHealth}
Here we note how the model laid out in section \ref{scn:modelMethods} of the paper generalises to a more flexible \citetappendix{Grossman2000}-style health model, where individual health is measured as a dynamic flow, depending on health at birth, as well as health in previous periods.  
Consider health at adulthood, when women are at risk of falling pregnant, and when one wishes to consider the interpretation of the RD model laid out in the paper, examining the intergenerational transmission of health at birth.  In this case, the analogue of health stocks of generation 1 laid out in Appendix \ref{sapp:singleGenModel} is:
\begin{equation}
    H_{it}=\phi_tH_i + \mu_tH_{i,t-1}(H_i,D_i,I_t^{post},\zeta_i) + \psi_tD_i + \varphi_tI_t^{post}(H_i,D_i,H_{i,t-1},\zeta_i)+ X_{it}\beta_t + \upsilon_{it},
\end{equation}
where now, $H_{it}$ is explicitly a function of $H_{i,t-1}$, and in turn, depends recursively on health in all previous periods: $H_{it}\equiv H_{it}(H_{i,t-1},H_i,D_i,I_t^{post},X)$.  We note two specific relevant considerations of this model of health stocks.  Firstly, it makes explicit that like parental investments, previous health stocks may depend upon initial health, treatment at birth, and subsequent stocks, as well as parental investment.  Importantly, this implies that $H_{it-1}$ may shift owing to threshold crossing, which must be taken into account when considering the interpretation of a reduced form RDD parameter during adulthood.  Secondly, we note that health stocks at birth are explicitly indicated as $H_i$, to note the lack of dependency on these response variables.

Now, as laid out in the paper, equations \ref{eqn:selec1a}-\ref{eqn:selec2} capture the implications of selection into fertility on the estimated intergenerational impacts of health investment at birth.  To see how this impacts the final decomposition of the estimated RDD parameter $\widehat\alpha$, we generalise the model here to incorporate $H_{i,t-1}$.  First, note that latent fertility now depends upon accumulated health stocks:
\begin{equation}
    \label{eqn:selec1A}
    \text{Early Fertility}^{*}_i = \gamma^E_1 H_i + \gamma^E_{2} D_{i} + \gamma^E_{3t}H_{i,t-1}(H_i,D_i,I_t^{post},\zeta_i) +\gamma^E_{4t}I_t^{post}(H_i,D_i,H_{i,t-1},\zeta_i) +  X_{it}\pi^E_t + \iota^E_{it},
\end{equation}
with a similar equation for Late Fertility.\footnote{For simplicity we are omitting the dependence on $B^{post}()$ in this section, though its inclusion would not affect the results here.} Similarly, the model of birth weight of generation 2 also depends upon the mother's accumulated health stocks: 
\begin{equation}
    \label{eqn:selec2A}
    BW_{ij} = H_{ij} + \phi_tH_i + \mu_tH_{i,t-1}(H_i,D_i,I_t^{post},\zeta_i) + \psi_tD_i + \varphi_tI_t^{post}(H_i,D_i,H_{i,t-1},\zeta_i)+ X_{it}\beta_t + u_{ij}, 
\end{equation}
with $BW_{ij}$ observed for individuals only if $\text{Early Fertility}_i=1$.  Following the model in the paper, the conditional expectation of birth weight is now:
\begin{eqnarray}
\label{eqn:HeckmanApp}
    E[BW_{ij}|\text{Early Fertility}_i=1] &=&  H_{ij} + \mu_tH_{i,t-1}(H_i,D_i,I_t^{post},\zeta_i) + \phi_tH_i + \nonumber \\ 
    && \psi_tD_i +  \varphi_tI_t^{post}(H_i,D_i,H_{i,t-1},\zeta_i)+X_{it}\beta_t +\nonumber \\ 
    && \rho_E\sigma\lambda_E[H_i,H_{i,t-1}(H_i,D_i,I_t^{post},\zeta_i),D_i,I^{post}(H_i,D_i,\zeta_i)]
\end{eqnarray}
where all notation follows section \ref{sscn:model} of the paper.  Finally, in this case, the extended interpretation of the RDD estimate $\widehat\alpha$ is now the following:
\begin{equation}
\label{eqn:intergenAlphaA}
\widehat\alpha = \psi_t\cdot \kappa + \mu_t\cdot \Delta H_{it-1}(c) + \varphi_t\cdot \Delta I_t^{post}(c)  + \rho_E\sigma \Delta \lambda_E[c,\Delta H_{it-1}(c),\Delta I_t^{post}(c)] + \rho_L\sigma \Delta \lambda_L[c,\Delta H_{it-1}(c),\Delta I_t^{post}(c)].
\end{equation}
There are two upshots from this extended model allowing for dynamics in mother's health.  Firstly, this opens up a direct new channel which may partially explain the RD estimate, which is that mother's health at $t-1$ (and recursively before that) impacts child's health at $t$, given that health investments at birth may directly impact mother's health at time $t$.  This is indicated by $\mu_t\cdot \Delta H_{it-1}(c)$.  Secondly, it additionally opens up a new sub-channel within the inverse Mills ratio, which is that a mother's health stock at time $t-1$ makes her more or less likely to give birth.  This is indicated by $\Delta H_{i,t-1}(c)$ within the inverse Mills ratio, where this is relevant if treatment receipt shifts later-life health around the cut-off $c$.  Note then that we can `close off' this channel if we can show that women are not more or less healthy at each time $t$ given treatment receipt.  
  
\subsection{Proceeding without a Normality Assumption}
\label{sapp:noHeckmanNormal}

In the main model laid out in the paper, selection into fertility was accounted for using the well-known \citetappendix{Heckman1974} correction, which relies on the assumption that the unobservables in the selection equations and the outcome equation are jointly normal. Under this assumption, selection enters the outcome equation through the inverse Mills ratio, which is a function of the estimated selection residuals. 

However, such an assumption is simply taken as it provides a convenient form for the selection equation.  It is, however, not required for the theoretical implications discussed at the end of Section \ref{sscn:model}.  To generalise this framework and relax the normality assumption, we can use a control function approach. Rather than relying on the inverse Mills ratio, we introduce a flexible correction term that accounts for selection in a more general way. Specifically, we replace the inverse Mills ratio with a nonparametric control function that depends on the estimated selection residuals. The modified expectations of second-generation birth weight conditional on selection are given by:
\begin{eqnarray}
  \label{eqn:CF_A}
  E[BW_{ij}|\text{Early fertility}_i=1]&=& H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(\cdot) + \varPsi B^{post}(\cdot) \nonumber \\ 
  &&+  CF_E\left(H_i,D_i,I^{post}(\cdot),B^{post}(\cdot),\widehat{\upsilon}^E\right),  \\
  \label{eqn:CF_B}
  E[BW_{ij}|\text{Later fertility}_i=1]&=& H_{ij} + \phi H_i + \psi D_i + \varphi I^{post}(\cdot) + \varPsi B^{post}(\cdot) \nonumber \\ 
  &&+  CF_L\left(H_i,D_i,I^{post}(\cdot),B^{post}(\cdot),\widehat{\upsilon}^L\right).
\end{eqnarray}

Here, $CF_E(\cdot)$ and $CF_L(\cdot)$ are unspecified control functions that capture the selection effects without imposing a specific functional form. These functions depend on the estimated residuals $\widehat{\upsilon}^E$ and $\widehat{\upsilon}^L$ from the first-stage selection equations. Unlike the inverse Mills ratio, which arises from the normality assumption, these control functions allow for a more flexible correction by incorporating potential nonlinearities or interactions.  This also would allow for IV-based procedures to be used in conjunction with control functions.

While this change the way that we write \eqref{eqn:HeckmanA}-\eqref{eqn:HeckmanB}, it does not alter the implications given that now our estimated effect from Section \ref{sscn:model} would be written as follows:
\begin{center}
\begin{tabular}{lll}
$\widehat\alpha\quad =$ &  $\psi\kappa$ & ``Structural''\\
 &  $+\quad \varphi\cdot\Delta I^{post}(c) + \varPsi\cdot\Delta B^{post}(c)$  & ``Behavioural''\\
 &  $+\quad \Delta CF_{E}\left[c,\Delta I^{post}(c),\Delta B^{post}(c),\Delta \widehat{\upsilon}^E(c)\right] \qquad\qquad$  & ``Selection (Early Fertility)''\\
& $+\quad \Delta CF_{L}\left[c,\Delta I^{post}(c),\Delta B^{post}(c),\Delta \widehat{\upsilon}^L(c)\right]$ & ``Selection (Later Fertility)''\\
\end{tabular}
\end{center}
Here, as in the model laid out in the paper, the key point is that any selection into fertility is relevant to the treatment effect if this selection is related to the outcome of interest, and any differential selection by age is similarly relevant if this is related to health outcomes at birth.

In the body of the paper, our interest in this model is as a way to consider potential channels discussed in the mechanism Section \ref{sscn:channels}.  We thus do not estimate the control function approach (or the Heckman selection approach laid out in the text), but rather evaluate whether evidence exists to support each of the channels posited by the model.  We nevertheless discuss the implications of Selection into treatment for the estimated parameter $\widehat\alpha$ in Section \ref{sscn:marginalVsAverage} of the text.

\clearpage


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\begin{landscape}
\section{Robustness Checks}
\label{app:robust}


\begin{figure}[htbp!]
    \caption{\textbf{RD Estimates of Intergenerational Impacts Varying Estimation Bandwidths}}
    \label{fig:BandwithVar32plus}
     \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_gweeks_bandwith_o32.eps}
      \caption{Gestation (weeks)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_peso_bandwith_O32.eps}
      \caption{Birth weight (grams)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_talla_bandwith_O32.eps}
      \caption{Birth length (cms.)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_deadata00_bandwith_O32.eps}
      \caption{Infant mortality}
    \end{subfigure}

    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_prem36_bandwith_O32.eps}
      \caption{Prematurity}
    \end{subfigure}   
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_vlbw_bandwith_O32.eps}
      \caption{Very low birth weight}
    \end{subfigure} 
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_fgrate_bandwith_O32.eps}
      \caption{Fetal growth rate}
    \end{subfigure} 
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_aindex_bandwith_O32.eps}
      \caption{Anderson index}
    \end{subfigure}
      \floatfoot{Notes: Each sub-plot portrays the impact of crossing the VLBW threshold on a specific second-generation health measure from Table \ref{tab:Gen2BirthOutcomes}, where the RDD specification is estimated within a bandwidth manually set to 90,100,110,\ldots,300 grams. Point estimates and 90\% and 95\% confidence intervals (blue and grey lines respectively) are shown using bias corrected estimates with heaping controls. The estimation sample consists of all children of first generation mothers who were born at 32 weeks or greater of gestation.}
  \end{figure}
\end{landscape}  

\begin{landscape}
    \begin{figure}[htbp!]
    \caption{\textbf{RD Estimates of Intergenerational Impacts Varying Estimation Bandwidths with Constant Relative Bias}}
    \label{fig:BiasBandwithConstant}
     \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_gweeks_biasbandwith_o32.eps}
      \caption{Gestation (weeks)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_peso_biasbandwith_O32.eps}
      \caption{Birth weight (grams)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_talla_biasbandwith_O32.eps}
      \caption{Birth length (cms.)}
    \end{subfigure}
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_deadata00_biasbandwith_O32.eps}
      \caption{Infant mortality}
    \end{subfigure}

    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_prem36_biasbandwith_O32.eps}
      \caption{Prematurity}
    \end{subfigure}   
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_vlbw_biasbandwith_O32.eps}
      \caption{Very low birth weight}
    \end{subfigure} 
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_fgrate_biasbandwith_O32.eps}
      \caption{Fetal growth rate}
    \end{subfigure} 
    \begin{subfigure}{.245\textwidth}
      \centering
      \includegraphics[width=0.98\linewidth]{figures/Appendix/g2_aindex_biasbandwith_O32.eps}
      \caption{Anderson index}
    \end{subfigure}
      \floatfoot{Notes: Each sub-plot portrays the impact of crossing the VLBW threshold on a specific second-generation health measure from Table \ref{tab:Gen2BirthOutcomes}, where the RDD specification is estimated within a bandwidth manually set to 90,100,110,\ldots,300 grams. Point estimates and 90\% and 95\% confidence intervals (blue and grey lines respectively) are shown using bias corrected estimates with heaping controls, where the bias correction calculation is conducted by holding a constant relative bias correction range to optimal bandwidth range as that used in principal models displayed in the paper. The estimation sample consists of all children of first generation mothers who were born at 32 weeks or greater of gestation.}
  \end{figure}
\end{landscape}  

\begin{landscape}
\begin{table}[htpb!]
    \centering
    \caption{\textbf{Alternative Specifications -- Gestational weeks, birth weight, birth length, infant mortality}}
    \label{tab:altspec1}
    \begin{tabular}{lcccccccc}\toprule
    &\multicolumn{4}{c}{Linear Running Variable} & \multicolumn{4}{c}{Quadratic Running Variable}   \\ \cmidrule(r){2-5}\cmidrule(r){6-9} 
    & Main & CERRD & Baseline & Bias Corrected & Main & CERRD & Baseline & Bias Corrected \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel A: Gestations weeks}} \\
    
    \input{tables/Appendix/_gwks.tex}
    
    \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel B: Birth weight (grams)}} \\
    
    \input{tables/Appendix/_peso.tex}
    
    \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel C: Birth Length (cms)}} \\
    
    \input{tables/Appendix/_talla.tex}
    
     \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel D: Infant Mortality}} \\
    
    \input{tables/Appendix/_deadata00.tex}
    
    \bottomrule
    \multicolumn{9}{p{19.6cm}}{{\footnotesize Notes: Alternative specifications are documented for each intergenerational outcome considered in Table \ref{tab:Gen2BirthOutcomes}. Each panel considers a specific outcome, based on local linear (columns 1-4), and local quadratic models (columns 5-8).  Alternative models consist of robust bias corrected models (column 1 and 5), models using \citetappendix{Calonicoetal2020a}'s minimum error coverage bandwidth selection (columns 2 and 4), using standard RD models without robust bias correction (columns 3 and 5), and using bias corrected but not robust models (columns 4 and 8).  Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}
\end{table}
\end{landscape}

\begin{landscape}
\begin{table}[htpb!]
    \centering
    \caption{\textbf{Alternative Specifications -- Prematurity, VLBW, Fetal Growth, Sex ratio, Anderson Index}}
    \label{tab:altspec2}
    \begin{tabular}{lcccccccc}\toprule
    &\multicolumn{4}{c}{Linear Running Variable} & \multicolumn{4}{c}{Quadratic Running Variable}   \\ \cmidrule(r){2-5}\cmidrule(r){6-9} 
    & Main & CERRD & Baseline & Bias Corrected & Main & CERRD & Baseline & Bias Corrected \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel E: Prematurity}} \\
    
    \input{tables/Appendix/_prem36.tex}
    
    \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel F: Very low birth weight status}} \\
    
    \input{tables/Appendix/_vlbw.tex}
    
    \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel G: Fetal growth rate}} \\
    
    \input{tables/Appendix/_fgrate.tex}

    
     \\ \midrule
    \multicolumn{9}{l}{\textbf{Panel I: Anderson Index}} \\
    
    \input{tables/Appendix/_aindex.tex}
    
    \bottomrule
    \multicolumn{9}{p{18.6cm}}{{\footnotesize Notes: Refer to notes to Table \ref{tab:altspec1}.  Identical models are presented, however considering alternative outcomes in each of panels E-I.  Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}
\end{table}
\end{landscape}

  \begin{table}[h!]
  \caption{\textbf{Identification Considerations -- Donut RD and Placebo Treatments}}
  \label{tab:robustness}
  \scalebox{0.86}{
  \begin{tabular}{lccccccc} \toprule
  & Gestation & Birth & Size & Infant & Premature & VLBW & Fetal \\ 
  & Length    & weight &     & Mortality &         &     & Growth \\
  & (1) & (2) & (3) & (4) & (5) & (6) & (7) \\ \midrule 
  \multicolumn{8}{l}{\emph{Panel A: Original Model}} \\
  \input{tables/Appendix/T6A_o32_hps}
  \multicolumn{8}{l}{\emph{Panel B: Donut RDD}} \\
      \input{tables/Appendix/T6B00_o32_hps}
      \input{tables/Appendix/T6B05_o32_hps}
      \input{tables/Appendix/T6B10_o32_hps}
      \input{tables/Appendix/T6B20_o32_hps}
  \multicolumn{8}{l}{\emph{Panel C: Alternative Cut-offs}} \\
      \input{tables/Appendix/T6C1250_o32_hps}
      \input{tables/Appendix/T6C1750_o32_hps}
      \input{tables/Appendix/T6C2000_o32_hps}
      \input{tables/Appendix/T6C2250_o32_hps}
      \input{tables/Appendix/T6C2500_o32_hps} \bottomrule
      \multicolumn{8}{p{16.5cm}}{{\footnotesize Notes: Panel A replicates estimates from Table \ref{tab:Gen2BirthOutcomes} for ease of comparison. Panel B estimates RDD models following \ref{eqn:baseline}, however removing observations within $x$ grams of the cut-off, where $x$ is indicated in the Donut hole value.  Panel C estimates RDD estimates following equation \ref{eqn:baseline}, however here rather than using the true discontinuity at 1,500 grams, alternative discontinuities at other maternal birth weight points are considered.  All additional notes follow those to Table \ref{tab:Gen2BirthOutcomes}.}}
  \end{tabular}}
\end{table}



\begin{table}[htpb!]
  \caption{\textbf{Intensive Health Investments and Birth Outcomes of the Second Generation (All births)}}
  \label{tab:Gen2BirthOutcomesALL}
  \scalebox{0.92}{
  \begin{tabular}{lcccc} \toprule
    & Gestation  & Birth weight 
    & Birth length & Infant  \\ 
    & (weeks) & (grams) & (cms) & Mortality \\
    \textbf{Panel A: Baseline Variables} & (1) & (2) & (3) & (4) \\
    \midrule
    \input{tables/Appendix/T2A_all_hps}
    \midrule
    & Prematurity & Very low  
    & Fetal growth  & Anderson \\
    & & birth weight & rate & Index \\
    \textbf{Panel B: Transformed Measures} & (5) & (6) & (7) & (8)  \\
    \midrule
    \input{tables/Appendix/T2B_all_hps}
    \bottomrule
    \multicolumn{5}{p{14.8cm}}{{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for mothers. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citetappendix{Calonicoetal2020a}.   Robust bias corrected standard errors are reported in parentheses.  Below standard errors, a one tailed t-test is calculated, which can be viewed as the support in favour of there actually being \emph{positive} intergenerational transmission to the second generation. q-sharpened p-values refer to corrections conducted across the entire class of outcomes.  
    * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
  \end{tabular}}
\end{table}

\begin{landscape}
\begin{figure}[htpb!]
  \caption{\textbf{Descriptive Plots of Parental Policy Receipt and Child Health Measures (All births)}}
  \label{fig:RDPlotGen2ALL}  
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_PESO.eps}
    \caption{Birth weight}
    \label{fig:BWRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_SEMANAS.eps}
    \caption{Gestational period}
  \label{fig:gestRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_TALLA.eps}
    \caption{Gestational length}
    \label{fig:sizeRDD}
\end{subfigure}

\newpage
%%%
\clearpage

\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_vlbw.eps}
    \caption{Very low birth weight}
    \label{fig:VLBWRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_premature.eps}
    \caption{Prematurity}
  \label{fig:premRDD}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
  \centering
  \includegraphics[width=1\linewidth]{figures/Appendix/figure2ALL_RDplot_fgrate.eps}
    \caption{Fetal growth rate}
    \label{fig:sgaRDD}
\end{subfigure}
\vspace{-2mm}
\floatfoot{Notes: Plots show separate quadratic fits estimated on each side of the 1,500 gram cut-off, in each case restricting attention to observations within the optimal bandwidth following \citet{Calonicoetal2014}.  95\% confidence intervals of the quadratic fit are estimated, and circles represent average outcomes in 20 gram bins.  Observations at 1,500 grams are plotted in binned averages, but are not used in estimating the quadratic fit.}
\end{figure}%
\end{landscape}%

\newpage
%%%
\clearpage
\begin{figure}[h!]
  \caption{\textbf{Distributional Impacts of Early Life Health Interventions on Second Generation Health Stocks (All births)}}
  \label{fig:distGen2All}  
  \begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.95\linewidth]{figures/Appendix/DistBW_all.pdf}
    \caption{Birth weight (All)}
    \label{fig:bwDistO32}
\end{subfigure}
  \begin{subfigure}{.49\textwidth}
  \centering
  \includegraphics[width=0.95\linewidth]{figures/Appendix/DistWeeks_all.pdf}
    \caption{Gestational length (All)}
    \label{fig:weeksDistO32}
  \end{subfigure}
\vspace{-4mm}
\floatfoot{Notes: Each point estimate (black square) and 90 and 95\% confidence interval (dark and light shaded areas respectively) refers to RDD estimates of the likelihood that a birth occurring to a treated girl born has health stocks (birth or gestational weight) below the cut-off indicated on the x-axis of each sub-plot.  Panels (a) and (c) refer to all individuals, while panel (b) and (d) refer to first generation individuals who were born at above 32 weeks of gestation, and are hence more clearly targeted by the policy.}
\end{figure}


\begin{figure}[h!]
  \caption{\textbf{Impacts of Early Life Health Interventions on  Fertility and Spontaneous Abortions (All births)}}
  \label{fig:fertAbortRDDALL}
   \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figures/Appendix/fertility_all.eps}
    \caption{Fertility}
    \label{fig:fertRDDALL}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figures/Appendix/fertility2_all.eps}
    \caption{Individual Has Given Birth}
    \label{fig:fertRDDALL}
    \end{subfigure}
    
   \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figures/Appendix/abortions_all.eps}
    \caption{Abortions}
    \label{fig:abortRDDALL}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=0.95\linewidth]{figures/Appendix/sexratio_all.eps}
    \caption{{Male Child}}
    \label{fig:abortRDDALL}
    \end{subfigure}
\vspace{-4mm}
 \floatfoot{Notes: Each point estimate and confidence interval refer to the impacts of early life medical investment on the number of births an individual has had by each age (panel (a)), the probability an individual has had a birth by each age (panel(b)), the number of abortions observed in hospitalisation data  (panel (c)), and the likelihood that an observed birth is a male (panel (d)). Thicker black error bars present 90\% CIs, while thinner error bars report 95\% CIs.  All estimates follow the procedures laid out in section \ref{scn:modelMethods}, and report RBC estimates using a local linear regression with a triangular kernel in the MSE optimal bandwidth.}
\end{figure}

\begin{figure}[htpb!]
  \caption{\textbf{Estimated RDD Effects by Second Generation Mother's Age}}
  \label{fig:ageEffectsRD}  
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/effectsByAge_PESO.eps}
    \caption{Birth weight}
    \label{fig:bwAge}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/effectsByAge_SEMANAS.eps}
    \caption{Gestational length}
    \label{fig:weeksAee}
  \end{subfigure}
\floatfoot{Notes: Each point estimate (black square) and 95\% confidence interval refer to RDD estimates of the effect of treatment on birth weight (panel (a)) and gestational length (panel (b)), for second generation mothers are ages indicated in panels. Each estimate is generated using the same sample and methods described in Notes to Table \ref{tab:Gen2BirthOutcomes}, however sub-setting exclusively to the ages indicated on the horizontal axis.}
\end{figure}

\begin{table}[ht!]
  \caption{\textbf{Intensive Health Investments and Birth Outcomes of the Second Generation in Adolescent Mothers (1992-2000 Cohorts)}}
  \label{tab:Gen2TeenBalance}
  \scalebox{0.99}{
  \begin{tabular}{lcccc} \toprule
    %& (1) & (2) & (3) & (4) \\
    & Gestation  & Birth weight 
    & Birth length & Infant   \\ 
    & (weeks) & (grams) & (cms) & Mortality \\
    \textbf{Panel A: Baseline Variables} & (1) & (2) & (3) & (4) \\
    \midrule
    \input{tables/Appendix/T2A_o32_cohort2000}
    \midrule
    & Prematurity & Very low  
    & Fetal growth  & Anderson \\
    & & birth weight & rate & Index \\
    \textbf{Panel B: Transformed Measures} & (5) & (6) & (7) & (8) \\
    \midrule
    \input{tables/Appendix/T2B_o32_cohort2000}
    \bottomrule
    \multicolumn{5}{p{14.8cm}}{{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for mothers. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}.  Robust bias corrected standard errors are reported in parentheses.  Below standard errors, a one tailed t-test is calculated, which can be viewed as the support in favour of there actually being \emph{positive} intergenerational transmission to the second generation. q-sharpened p-values refer to corrections conducted across the entire class of outcomes. 
    * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
  \end{tabular}}
\end{table}


\begin{table}[h!]
  \caption{\textbf{Effects Conditioning on Education and Partnership Choices}}
  \label{tab:condEducPartners}
  \scalebox{0.86}{
  \begin{tabular}{lcccccccc} \toprule
  & Gestation & Birth & Size & Infant & Premature & VLBW & Fetal  & Anderson \\
  & Length    & weight &     & Mortality &         &     & Growth & Index \\
  & (1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) \\ \midrule
  \multicolumn{9}{l}{\emph{Panel A: Controlling for Education}} \\
  \input{tables/Appendix/T2A_o32_controls}\\
  \multicolumn{9}{l}{\emph{Panel B: Conditioning on Highly Educated}} \\
  \input{tables/Appendix/T2A_o32_highEd}\\
  \multicolumn{9}{l}{\emph{Panel C: Controlling for partner presence}} \\
  \input{tables/Appendix/T2A_o32_control_relationship}\\
  \multicolumn{9}{l}{\emph{Panel D: Conditioning on partner presence}} \\
  \input{tables/Appendix/T2A_o32_relationship}
  \bottomrule
  \multicolumn{9}{p{18.2cm}}{{\footnotesize Notes: Results replicate estimates from Table \ref{tab:Gen2BirthOutcomes} conditioning on mother's education at time of birth (panel A), or whether the father was observed to be present in the child's registration (panel C).  Results condition on the mother coming from a highly educated (grandmother's education above the median) (panel B), or on the father being present in the child's registration (panel D).  All additional notes follow those to Table \ref{tab:Gen2BirthOutcomes}.}}
  \end{tabular}}
\end{table}

\clearpage

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\section{Additional Details: Selection into Treatment}
\label{app:selection}
\subsection{Selection into Treatment in the First Generation}
\label{app:2dim}
In the body of the paper, we focus on selection into second generation births which owe to selective fertility.  However, an identical issue could arise if selection occurs due to survival at birth when intensive medical treatments were initially provided.  As documented by \citet{Bharadwajetal2013}, and as discussed further in Appendix \ref{scn:resultsGen1}, this policy resulted in selective survival of individuals who received more intensive treatment, and as such, likely saved individuals with poorer health stocks.  In order to consider whether this initial selective survival can explain our observed results, we conduct a counterfactual experiment where we impute non-surviving individuals on the right hand side of the cut-off, and consider whether specific life courses for these individuals could explain away the negative observed intergenerational transmission.

The results of this exercise are provided in Table \ref{tab:counterfactuals}.  This table makes clear that while selective survival of the first generation could affect the magnitude of the estimates, it would generally be insufficient to invert the sign of point estimates.  For example, consider the case of child birth weight.  If all counterfactual individuals on the right-hand side of the cut-off had a birth weight equal to that of the tenth percentile, the intergenerational effect could be as small as -163 grams.  While if these non-surviving individuals had birth weight in the 90\textsuperscript{th} percentile, the true effect could be as large as -281 grams.  This exercise is particularly clear in the case of a child's low birth weight status.  Regardless of how extreme we make the counterfactual scenarios, the true effect remains large, and statistically significant, even at 1\%.  

This analysis considers counterfactual stocks of children who did not exist in the second generation due to selective survival in the first generation.  More generally, it is illustrative to consider the range of counterfactual scenarios over which it would be sufficient to turn negative observed intergenerational impacts into positive effects.  In Table \ref{tab:counterfactuals} we assume that all imputed individuals would have had fertility rates which were equal to those of surviving individuals in their birth weight bin.  

Specifically, the exercise consists of the following.  Using estimates of the policy impact on infant mortality, we calculate the number of individuals on the right-hand side of the birth weight cut-off who we estimate would have survived had they received the treatment.  This is estimated precisely using the RDD models discussed in this paper.  We then impute the proportion of these individuals who would have given birth by each age using 20 gram birth weight bin-specific actual averages observed of this variable.  We then add these additional `counterfactual' births to our second generation sample (note that given different optimal bandwidths, the number of imputed individuals will vary slightly by outcome).  As we do not know what their health outcomes would have been at birth, we consider a range of scenarios, imputing health outcomes across the actual distribution of health at birth.

The results from this activity are plotted for birth weight (Figure \ref{fig:select2d}a), gestational weeks (Figure \ref{fig:select2d}b), size at birth (Figure \ref{fig:select2d}c), and fetal growth rate (Figure \ref{fig:select2d}d).  Binary outcomes are not considered, as these often entirely omit extreme outcomes such as VLBW which occur in less than 5\% of cases.  Considering birth weight as the outcome of interest, it appears highly unlikely that selective survival at birth could explain the observed negative intergenerational transfers from mothers to children.  For this to be the case, non-surviving individuals would have had to be highly fertile (each having 1 birth), while at the same time giving birth to children with very low health stocks (all at the 5\textsuperscript{th} health percentile).  Similar extreme patterns are observed for both gestational weeks and the fetal growth rate.  In the case of size at birth, more feasible counterfactual outcomes exist which could explain away the negative intergenerational transmission (for example, average birth rates, and all counterfactual babies being born at the 20\textsuperscript{th} health percentile or below.  Nevertheless, across all outcomes considered, the evidence broadly suggests that correcting for survival at birth in the first generation would not be sufficient to turn around observed results in all health dimensions.


\subsection{Compositional Effects and Fertility Selection}
\label{sapp:selectionMore}
In the body of the paper we discuss a number of additional tests related to fertility and selection, and these are collected in this Appendix.  These results are summarised below:
\begin{itemize}
\item Table \ref{tab:fertBirths}: Reports the RD estimate where the outcome is the number of children born. 
\item Table \ref{tab:fertilitySelectionRDD}: Corresponds to formal RDD estimates which examine whether the characteristics of parents of second generation children differ across the treatment threshold.  These correspond to descriptive results presented in Figure \ref{scn:selection} of the main paper.
\item Figure \ref{fig:ageEffects}: Reports marginal effects of health at birth by mother's age from descriptive regressions (left-hand panel) and mother fixed effect models (right-hand panels).  Models consist of estimating:
\[
Y_{ij} =  \mu_{age} + \phi_j + \varepsilon_{ij}
\]
for child $i$ born to mother $j$, where $\mu_{age}$ refers to mother age at birth fixed effects, and $\phi_j$ refers to mother fixed effects (included in right-hand column, but not left-hand column). Means along with their 99\% confidence intervals are shown for each age fixed effect.
\item Figure \ref{fig:fertDesc}: Plots showing descriptive quantities for individuals marginally to the left (treated) and marginally to the right (untreated) of the cut-off.  These show the mean number of children born by age (panel (a)), the probability of birth at a given age (panel (b)), whether an individual has had any births by a given age (panel (c)), and whether an individual has a birth at a given age (panel (d)).  The difference between panels (a)/(c) and (b)/(d) is that the prior reports values \textit{by} age (\textit{i.e.}\ is cumulative), while the latter reports values \textit{at} age (\textit{i.e.}\ is not cumulative).
\item Figure \ref{fig:fertDescMed}: Reports an identical plot to Figure \ref{fig:fertDesc}, however around the median rather than around the treatment cut-off.
\item Figure \ref{fig:ageEffectsCharac}: Reports similar effects to those in Figure \ref{fig:ageEffects}, but estimating separately by characteristics of exposed mothers or their families.
\item Table \ref{tab:fertParentLabour}: Balance tests where indicators for an individuals age at birth are regressed on treatment receipt within the bandwidth close to 1,500 grams ($\pm$ 245.5 grams, as per Table \ref{tab:Gen2BirthOutcomes}).
\end{itemize}
  
\subsection{Additional Details: Marginal versus Average Effects in an RDD with Selection}
\label{sapp:selectionMA}
In Section \ref{sscn:marginalVsAverage} we discuss what can be said about direct (`structural') effects of treatment receipt on health at birth of the following generation, and lay out a decomposition of estimate treatment effects in \eqref{eqn:RDselection}. To see how we arrive at \eqref{eqn:RDselection}, note that if we have two groups (the marginal mothers and the always mothers, we can write our estimated treatment effect as follows: 
\begin{eqnarray}
\alpha&=&\left(E[Y^A_{i}|BW<1500]-E[Y^A_{i}|BW\geq 1500]\right)(1-\mu) \nonumber  \\
&&+\left(E[Y^M_{i}|BW<1500]-E[Y^A_{i}|BW\geq 1500]\right)\mu
\end{eqnarray}
where $\mu$ refers to the portion of marginal mothers and $(1-\mu)$ the remaining portion of always-mothers. The key point to note here is that \textit{because we do not observe} marginal mothers on the untreated side of the cut-off, we use the outcome among always mothers as the comparison group in both cases, and this is why $E[Y^A_{i}|BW\geq 1500]$ appears two times.

Now, if we add and subtract $\mu E[Y^M_{i}|BW<1500]$, and rearrange, we can write the estimated  treatment effect as follows:
\begin{eqnarray}
\label{eqn:decompAppend}
\alpha&=&\left(E[Y^A_{i}|BW<1500]-E[Y^A_{i}|BW\geq 1500]\right)(1-\mu) \nonumber  \\
&&+\left(E[Y^M_{i}|BW<1500]-E[Y^M_{i}|BW\geq 1500]\right)\mu \nonumber \\
&&+\left(E[Y^M_{i}|BW\geq 1500]-E[Y^A_{i}|BW\geq 1500]\right)\mu. 
\end{eqnarray}
In the paper we refer to the quantity $E[Y^M_{i}|BW\geq 1500]-E[Y^A_{i}|BW\geq 1500]$ as $\Delta$ (the difference between births occurring to marginal and always treated mothers in the absence of treatment).  Thus, we can re-write \eqref{eqn:decompAppend} as:
\begin{eqnarray}
\label{eqn:decompAppend}
\alpha&=&\alpha^A(1-\mu) +  \alpha^M\mu + \Delta\mu, 
\end{eqnarray}
which is the decomposition provided in the paper.

We present a graphical presentation of this decomposition in Figure \ref{fig:RDselection}.  This also makes clear why an RDD estimate with selection may be negative even if the direct policy effect is positive in both groups.  Namely, as laid out in Figure \ref{fig:RDselection}, if we do not observe outcomes for the selected group on one side of the cut-off (as indicated by the dashed blue line), and if these are instead compared to outcomes for a group which is positively selected, our estimated effect may be substantially different to a weighted average of the direct structural effects in each group.

\clearpage


\begin{table}[h!]
  \caption{{\textbf{Selective survival and second generation outcomes -- counterfactual analysis}}}
  \label{tab:counterfactuals}
  \scalebox{0.86}{
   \begin{tabular}{lcccccc} \toprule
    & & \multicolumn{5}{c}{Counterfactual percentile assumed for survival} \\ \cmidrule(r){3-7}
    & Baseline & 10\textsuperscript{th} & 30\textsuperscript{th} & 50\textsuperscript{th} & 70\textsuperscript{th} & 90\textsuperscript{th}  \\ \midrule
    \multicolumn{7}{l}{\textit{Panel A: Child's gestational length}}\\
    \input{tables/Appendix/select_gwks.tex}
    \midrule
    \multicolumn{7}{l}{\textit{Panel B: Child's birth weight}}\\
    \input{tables/Appendix/select_peso.tex}
    \midrule
    \textit{Panel C: Child's birth size} &&&&&&\\
    \input{tables/Appendix/select_talla.tex}
    \midrule
    \multicolumn{7}{l}{\textit{Panel D: Child's infant mortality}}\\
    \input{tables/Appendix/select_deadata00.tex}
    \midrule
    \multicolumn{7}{l}{\textit{Panel E: Child premature}}\\
    \input{tables/Appendix/select_prem36.tex}
    \midrule
    \multicolumn{7}{l}{\textit{Panel F: Child VLBW}}\\
    \input{tables/Appendix/select_vlbw.tex}
    \midrule
    \multicolumn{7}{l}{\textit{Panel G: Fetal growth rate}}\\
    \input{tables/Appendix/select_fgrate.tex}
    \bottomrule
    \multicolumn{7}{p{16.7cm}}{\footnotesize Notes: Each panel displays outcomes under different counterfactual assumptions related to future outcomes for individuals who selectively did not survive birth as response to not receiving intensive treatment in generation 1. The left-hand panel replicates original estimates of policy receipt on intergenerational outcomes, and then additional columns impute outcomes for observations who selectively did not survive birth on the right-hand side of the treatment cut-off, assuming counterfactual outcomes at different percentiles of the health distribution at birth.}
    \end{tabular}}
\end{table}



\begin{figure}[ht!]
  \caption{{\textbf{Intergenerational Transmission Under Alternative Health and Fertility Counterfactuals}}} 
  \label{fig:select2d}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/peso100.eps}
    \caption{Birth weight}
    \label{fig:selectionBW}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix//gwks100.eps}
    \caption{Gestation Weeks}
    \label{fig:selectionBW}
  \end{subfigure}
  
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/talla100.eps}
    \caption{Size at birth}
    \label{fig:selectionBW}
  \end{subfigure}
  \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=1\linewidth]{figures/Appendix/fgrate100.eps}
    \caption{Fetal growth rate}
    \label{fig:selectionBW}
  \end{subfigure} 
  \floatfoot{Notes: Each point represents estimates of the impact of intensive health receipt at first generation births on the second generation health outcomes indicated as plot labels.  Each estimate comes from a separate RD model based on all surviving observations, as well as imputed outcomes for individuals estimated to have not survived on the right-hand side of the birth weight cut-off due to lack of policy receipt.  Imputations are made varying health outcomes (indicated on the horizontal axis), and fertility (indicated in the legend).  All estimation details of each RD model follow those described in Table \ref{tab:Gen2BirthOutcomes}.}
\end{figure}



\begin{table}[htpb!]
  \caption{{\textbf{Impacts of Early Life Health Interventions on Number of Children}}}
  \label{tab:fertBirths}
  \scalebox{0.85}{
  \begin{tabular}{lc} \toprule
    \input{tables/Appendix/TA18_o32} 
    \bottomrule
    \multicolumn{2}{p{7cm}}{{\footnotesize Notes: The table displays estimates of the change in the number of births from above to below the 1,500 gram assignment threshold. Local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}. Robust bias corrected standard errors are reported in parentheses. * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}}
\end{table}



\begin{landscape}
\begin{table}[htpb!]
  \caption{{\textbf{Is there Policy-Driven Selection into Childbirth? RD Estimates}}}
  \label{tab:fertilitySelectionRDD}
  \scalebox{0.95}{
  \begin{tabular}{lcccccccccc} \toprule
    &\multicolumn{3}{c}{Mother} & \multicolumn{3}{c}{Father} &  &{Parent's} & Observed & Urban\\ \cmidrule(r){2-4}\cmidrule(r){5-7}
    & Age & Education & {Employed} & Age & Education & {Employed} & Married & {Age Diff.}\ & Dad & Status\\ \midrule
    \input{tables/Appendix/fig7_o32_hps}
    \bottomrule
    \multicolumn{11}{p{22.6cm}}{\footnotesize Notes: Each column displays estimates of the change in the given dependent variable from above to below the 1,500 gram assignment threshold for newborns. In each case, local linear regression is used with a triangular kernel, calculating the MSE optimal bandwidth of \citet{Calonicoetal2020a}. Robust bias corrected standard errors clustered at the gram level are reported in parentheses. p-values for one-sided tests are shown in square brackets, where in each case the null hypothesis is that there is a negative policy impact. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}
    \end{tabular}}
\end{table}
\end{landscape}








\begin{figure}
  \caption{{\textbf{Weight at Birth by Mothers' Age}}}
  \label{fig:ageEffects}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/birthWeightAge.pdf}
    \caption{Maternal age and child's birth weight}
    \label{fig:ageEffectsBW}
  \end{subfigure}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/birthWeightAge_FE.pdf}
    \caption{Maternal age and child's birth weight -- Mother FE}
    \label{fig:ageEffectsBW_FE}
  \end{subfigure}%
  
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/vlbwAge.pdf}
    \caption{Maternal age and child's VLBW status}
    \label{fig:ageEffectsVLBW}
  \end{subfigure}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/vlbwAge_FE.pdf}
    \caption{Maternal age and child's VLBW status -- Mother FE}
    \label{fig:ageEffectsVLBW_FE}
  \end{subfigure}%

  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/prematureAge.pdf}
    \caption{Maternal age and child's prematurity}
    \label{fig:ageEffectsPremature}
  \end{subfigure}
  \begin{subfigure}{0.49\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/prematureAge_FE.pdf}
    \caption{Maternal age and child's prematurity -- Mother FE}
    \label{fig:ageEffectsPremature_FE}
  \end{subfigure}%
  \floatfoot{Notes: Each sub-plot documents average outcomes of individuals born to mothers whose age is indicated on the horizontal axis.  Models are estimated with all observations in our period under study and include age fixed effects descriptively (left panel), or age fixed effects in a mother fixed effect model (right panel). Panels (a) and (b) consider the birth weight of children, panels (c) and (d) consider the average proportion of children who are very low birth weight (weight $<$ 1,500 grams), while panels (e) and (f) consider the average proportion of children who are premature (gestation $<$ 37 weeks). In the case of mother fixed effect models, marginal effects are documented for each age fixed effect.}
\end{figure}

\begin{figure}[h!]
  \caption{\textbf{{Childbirth Around the 1,500 Grams Threshold}}}
  \label{fig:fertDesc}
    \centering
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/avgbirthsby_vlbw.eps}
    \caption{Average childbirths by age}
    \label{fig:pfert_a}
    \end{subfigure}%
     \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/anybirthsat_vlbw.eps}
    \caption{Probability of childbirth at age}
    \label{fig:pfert_b}
    \end{subfigure}%
    
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/anybirthsby_vlbw.eps}
    \caption{Probability of childbirth by age}
    \label{fig:pfert_c}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/firstbirthat_vlbw.eps}
     \caption{Probability of first childbirth at age}
    \label{fig:pfert_d}
    \end{subfigure}%
\vspace{-4mm}
 \floatfoot{Notes:  Each point shows the estimated average number of childbirths by the age indicated (panel (a)), the probability that a mother has a birth at the age indicated (panel (b)) and by the age indicated (panel (c)), and the probability that a mother has the first birth at the age indicated (panel (d)). This is relative to the 1,500 grams cut-off and based on the 134.4 gram optimal bandwidth cut-off when considering infant mortality for the first generation.}
\end{figure}

\begin{figure}[h!]
  \caption{\textbf{{Probability of Childbirth Relative to Median Birth Weight}}}
  \label{fig:fertDescMed}
    \centering
       \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/avgbirthsby_median.eps}
    \caption{Average childbirths by age}
    \label{fig:pfert_a}
    \end{subfigure}%
     \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/anybirthsat_median.eps}
    \caption{Probability of childbirth at age}
    \label{fig:pfert_b}
    \end{subfigure}%
    
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/anybirthsby_median.eps}
    \caption{Probability of childbirth by age}
    \label{fig:pfert_c}
    \end{subfigure}
    \begin{subfigure}{.49\textwidth}
    \centering
    \includegraphics[width=\linewidth]{figures/Appendix/firstbirthat_median.eps}
     \caption{Probability of first childbirth at age}
    \label{fig:pfert_d}
    \end{subfigure}%
\vspace{-4mm}
 \floatfoot{Notes: Each point shows the estimated average number of childbirths by the age indicated (panel (a)), the probability that a mother has a birth at the age indicated (panel (b)) and by the age indicated (panel (c)), and the probability that a mother has the first birth at the age indicated (panel (d)). This is relative to the median of 3,300 grams.}
\end{figure}





\begin{landscape}
\begin{figure}
  \caption{{\textbf{Weight at Birth by Mother's Age and Parent Characteristics}}}
  \label{fig:ageEffectsCharac}
  \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/birthWeightAge_educ_FE.pdf}
    \caption{Maternal age and children's birth weight}
    \label{fig:ageEffectsEdBW_FE}
  \end{subfigure}
  \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/vlbwAge_educ_FE.pdf}
    \caption{Maternal age and children's very low birth weight}
    \label{fig:ageEffectsEdVLBW_FE}
  \end{subfigure}
  \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/prematureAge_educ_FE.pdf}
    \caption{Maternal age and children's premature}
    \label{fig:ageEffectsEdPremature_FE}
  \end{subfigure}%

    \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/birthWeightAge_agediff_FE.pdf}
    \caption{Maternal age and children's birth weight}
    \label{fig:ageEffectsPartnerBW_FE}
  \end{subfigure}
    \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/vlbwAge_agediff_FE.pdf}
    \caption{Maternal age and children's very low birth weight}
    \label{fig:ageEffectsPartnerVLBW_FE}
  \end{subfigure}
  \begin{subfigure}{0.33\textwidth}
    \centering
    \includegraphics[width=0.99\linewidth]{figures/Appendix/prematureAge_agediff_FE.pdf}
    \caption{Maternal age and children's premature}
    \label{fig:ageEffectsPartnerPremature_FE}
  \end{subfigure}%
  \floatfoot{Notes: Each sub-plot documents by group the  outcomes of individuals born to mothers whose age is indicated on the horizontal axis including mother-level fixed effects. Panels (a) and (d) consider the birth weight of second-generation children, panels (b) and (e) consider the average proportion of second-generation children who are very low birth weight (weight $<$ 1,500 grams), while panels (c) and (f) consider the average proportion of second-generation children who are premature (gestation $<$ 37 weeks). Optimally spaced bins and their 99\%  confidence intervals are documented as black points and error bands.}
\end{figure}
\end{landscape}

\newpage
\begin{table}[ht!]
  \caption{{\textbf{Mother's Age Baseline Test by Family Characteristics -- Generation 2}}}
  \label{tab:fertParentLabour}
  \scalebox{0.75}{
  \begin{tabular}{lccccccccc} \toprule
    & &\multicolumn{2}{c}{Mother's Education} &\multicolumn{2}{c}{Grandmother's Education} &\multicolumn{2}{c}{Parents' Age Difference} &\multicolumn{2}{c}{Observed Father}   \\
    \cmidrule(r){3-4}\cmidrule(r){5-6}\cmidrule(r){7-8}\cmidrule(r){9-10}
    &    & Below median &   At or above          & Below median  &     At or above       &  Up to   & More than&         & Non\\ 
    &All & (12 years)   & median & (9 years)    & median  & 5 years & 5 years & Observed&Observed \\ \midrule
    \input{tables/Appendix/TA20_y15_o32} 
    \input{tables/Appendix/TA20_y16_o32} 
    \input{tables/Appendix/TA20_y17_o32} 
    \input{tables/Appendix/TA20_y18_o32} 
    \input{tables/Appendix/TA20_y19_o32} 
    \input{tables/Appendix/TA20_y20_o32} 
    \input{tables/Appendix/TA20_y21_o32} 
    \input{tables/Appendix/TA20_y22_o32}
    \input{tables/Appendix/TA20_y23_o32} 
    \input{tables/Appendix/TA20_y24_o32} 
    \input{tables/Appendix/TA20_y25_o32} 
    \bottomrule
    \multicolumn{10}{p{21.5cm}}{{\footnotesize Notes: Each column displays estimates of the change in the probability of a mother gives birth at a certain age from above to below the 1,500 gram assignment threshold in the given sub-sample. Significance stars for two-sided test: * p$<$0.10; ** p$<$0.05; *** p$<$0.01.}}
    \end{tabular}}
\end{table}




\begin{figure}
\caption{Understanding Regression Discontinuity Estimates With Sample Selection}
\label{fig:RDselection}
\begin{tikzpicture}
\begin{axis}[
    axis lines = middle,
    enlargelimits,
    xmin=-4, xmax=4,
    ymin=-0.5, ymax=15,
    xlabel={Birth weight (G1)},
    ylabel={Health at birth (G2)},
    domain=-5:10,
    samples=100,
    legend pos=north west,
    width=15cm, height=9cm,
    xtick={0}, % Only tick at x = 0
    xticklabels={1500}, % Label 1500 at x = 0
    ytick=\empty, % Remove all y-axis ticks axis line style={none} % Remove all axis lines
 ]
% First wavy line for the left side (group 1)
\addplot[
    blue,
    ultra thick,
    domain=-4:0
] {sin(deg(x)) + 7}; % Keeps it off zero on the y-axis

% Second wavy line for the left side (group 2)
\addplot[
    red,
    ultra thick,
    domain=-4:0
] {0.8*sin(deg(x)) + 12.5}; % Different sine function for variety

% First wavy line for the right side (group 1 with higher jump)
\addplot[
    blue,
    ultra thick,
    dashed, % Dashed line
    domain=0:4
] {0.8*sin(deg(x)) + 5}; % A different function, still with higher jump

% Second wavy line for the right side (group 2 with lower jump)
\addplot[
    red,
    ultra thick,
    domain=0:4
] {sin(deg(x)) + 10}; % Different sine function, with lower jump


\node at (axis cs: 0.5, -1) {$1500$ g};


% Add curly braces to indicate the jumps
\draw[decorate, decoration={brace,amplitude=10pt,mirror}] (axis cs:0,7) -- (axis cs:0,5) node[midway,xshift=-15pt] {$\tau^m$};
\draw[decorate, decoration={brace,amplitude=10pt,mirror}] (axis cs:0,12.5) -- (axis cs:0,10) node[midway,xshift=-15pt] {$\tau^a$};


\legend{Marginal mothers, Always mothers}
\end{axis}
\end{tikzpicture}
\end{figure}


\clearpage



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\section{A Back-of-the-Envelope Estimation of Policy Costs and Benefits}
\label{sscn:discussion}
%Read  to do this...
Results from sections \ref{sscn:resultgen2}-\ref{sscn:channels} suggest that the marginal impacts of this intensive early life medical care may actually act to \emph{increase} costs levied in the second generation.  This is at odds with the short-term marginal returns documented in previous literature pointing to greater survival \citepappendix{Almondetal2010}, increased educational performance \citepappendix{Bharadwajetal2013}, and reduced links to state support programmes \citepappendix{Chynetal2021}.  It is illustrative to consider what these second generation results imply vis-à-vis the policy's cost, and positive short term returns.  The importance of considering such life-cycle returns to social policies has been espoused in \citetappendix{Garciaetal2020}, and here we can move beyond the life cycle of the first generation, into second generation outcomes.

To do this, we consider estimates of the initial costs of providing marginal medical care to treated individuals, the estimated present value of first generation benefits of policy receipt, and the estimated present value of second generation costs.  We incorporate into these calculations the welfare cost of financing taxation, and consider a money metric to compare across different policy domains.  In each case, we are fortunate to have a directly matched control group to the treated group of individuals impacted by the policy. Namely, the counterfactual outcome for children treated at birth are children who narrowly miss out given the birth weight assignment rule.  Nevertheless, we note that this is simply a back-of-the envelope activity, principally to allow us to determine how important the negative intergenerational transmission observed here may be compared with first generation benefits. Below we lay out estimated present values for costs and benefits along with any necessary assumptions to arrive to these figures (calculations are summarised in Appendix Table \ref{tab:CBA}).  This exercise is conducted from the point of view of a single marginally treated first generation child who is born in a public hospital, starting from the age of 0, with all costs and benefits discounted to the present day (and expressed in terms of real costs in 2022).  In closing this sub-section, we note a number of elements which are likely affected by the initial medical care receipt, but which are not feasible to reduce to a money metric, and which are hence omitted from the calculation of policy returns.

The most obvious programme cost which must be covered by public funds are the costs of initial medical care.  In the US, \citetappendix{Almondetal2010} estimate that the cost of marginal medical care provided at this cut-off is 9,450 USD.  In order to consider the costs in the context of Chile, we calculate the full costs associated with marginal estimated changes in hospitalisation.  The additional use of hospitalisation days is documented in Figure \ref{fig:fertRDHosp}, at approximately 3.5 days in the first year of life, 1.2 days in year 2, 0.7 in year 3 and 0.5 in year 4 (after which no marginal changes in hospitalisation days are observed). Figures estimated from Chile related to the \emph{total} cost of intensive care days (including medical inputs and care) suggest a value of 480,047 CLP in 2011 \citepappendix{Alvearetal2013}, or approximately 694,000 CLP in current terms, which is equivalent to 855 USD per day.\footnote{Note that \citetappendix{Almondetal2010} estimate a cost of marginal treatment at birth of 9,450 USD in hospital costs in 2011 USD.  Here, based on our estimate of 3.6 additional days of hospitalisation at birth, the equivalent instantaneous cost in USD of hospitalisation in Chile is  $855\times3.6= 3078$.  Even when incorporating the welfare cost of taxation suggested by \citetappendix{Garciaetal2020}, this value corresponds to 4,617 in 2022 USD, or 3,661 in 2011 USD.  Thus costs in Chile are around a third of those in the US, in line with relative cost and hospitalisation usage indexes in the two countries \citepappendix{OECD2017}.}  Taking the net present value of costs to the health system at the time of birth, discounted at a 5\% discount rate, this gives expected costs of 4,649 USD over the first years of the child's life.  If we additionally incorporate the welfare cost of taxation noting the deadweight loss associated with the collection of tax revenues \citepappendix{Feldstein1999}, suggested by \citetappendix{Garciaetal2020} as 50 cents on the dollar, this suggests an initial policy cost of 6,974 USD.

% At moment of birth t=0: 3.5*855*(1.05)^(-1)+1.5*855*(1.05)^(-2)+0.8*855*(1.05)^(-3)+0.5*855*(1.05)^(-4)

% UF Jan 1 2011: 21.456,25
% UF Jan 1 2022: 30.996,73 
% hospital day 2022: 480047*(30996.73/21456.25)

%US CPI in 2011	224.939
%US CPI in 2022	283.716

Early medical life receipt has been documented to be associated with a number of benefits.  We consider here a reduced metric of first generation labour market outcomes, mapping from educational benefits documented by \citetappendix{Bharadwajetal2013}.  Based on their estimated impacts of policy receipt on education in the first generation, and a back of the envelope calculation of the labour market returns to education, they suggest that marginal receipt of intensive health investments at birth may increase incomes by 2.7\%.   If we discount the expected flow of future earnings back to birth, based on the median income in Chile in 2021, this suggests directly attributable changes in salaries of 980 USD.  Thus, labour market concerns alone do not cover the initial outlay, but if one additionally accounts for the value of a statistical life (VSL), and the marginal likelihood that individuals die during their first year of life, this suggests substantial additional benefits of 91,760 USD \citepappendix{mardones_riquelme_2018}.
%3.7 million US*0.0386 change in IMR

In addition, we consider the associated costs to the reduction in health at birth of the second generation. We observed three main cost sources: an increase in the probability of receiving treatment in the second generation, an increase in infant mortality, and a decrease in birth weight.  We calculate the net present value of the costs and benefits at the time of the mother's birth considering 0.24 additional births up to 25 years documented in Figure \ref{fig:fertAbortRDD} for those mothers who were initially treated and the welfare cost of financing taxation.  Taking the increase of 0.045 in the probability of being born at less than 1,500 grams and therefore of receiving the treatment, the net present value of the initial medical care costs is 22 USD.  Due to the increase of 0.006 observed in the infant mortality rate for those born to treated mothers, a cost of 1,573 USD associated with the value of statistical lives lost must also be included. Finally, we consider changes in future income through changes in schooling.  Educational benefits documented by \citetappendix{BehrmanRosenzweig2018} indicate that increasing the birth weight by 1 lb.\ increases adult earnings by more than 7\%.  Thus, the 217 gram decrease in birth weight is associated with an additional cost of 86 USD.  The above suggests a second generation cost of the policy of 1,682 USD.  In contrast, we observe a decrease in hospitalisations during the first years of life which corresponds to an expected second generation benefit of 60 USD.

These calculations suggest that the total estimated cost of the policy amounts to 8,656 USD while the estimated benefit of the policy is 92,800 USD. If we only consider the 980 USD corresponding to labour returns to education received by the first generation and the 60 USD due to the decrease in early hospitalisation for the second generation, the amount is not enough to compensate the initial costs of medical care and the reduction in health of the second generation. However, when incorporating the marginal willingness to pay for the reduction of infant mortality through the VSL, the benefits are around 15 times the total costs.  It is important to note that while we can provide a back-of-the-envelope estimate of the policy's net present value, this is necessarily based on a partial picture of the policy's full set of benefits.  While we can feasibly value the policy's impact on health, education and future labour market outcomes, there are a number of benefits which we cannot easily quantify.  This includes factors such as reduced rates of bereavement given lower rates of infant mortality and potential lower rates of pregnancy loss in the second generation,\footnote{Both of these events have obvious and considerable costs to well-being over myriad dimensions \citepappendix{Rogersetal2008,Ogwuluetal2015,PerssonRossinSlater2018}.  However adequately capturing their true value would be difficult.} as well as the policy's impact on allowing women and families greater autonomy to achieve their desired fertility.  

It is important to note that due to a lack of recorded father's identifier, our estimated second generation costs and benefits are all based on mother--child links.  In general, there is evidence to suggest significant intergenerational links between fathers and their children \citepappendix[for example]{Giuntellaetal2019}.  The calculations here all refer to costs and benefits for a single individual, and so are not affected if impacts are identical for fathers and mothers.  The literature suggests that links between fathers and their children are generally smaller than those between mothers and their children \citep{Changetal2024}, which, if anything, suggests that second generation costs and benefits would be lower for fathers. Given that costs exceed benefits in the second generation, all told it seems that the inclusion of fathers would only increase the total present value of policy receipt. 
\begin{table}[htpb!]
    \centering
    \caption{Present value of costs and benefits of policy}
    \label{tab:CBA}
    \begin{tabular}{lcc} \toprule
    & USD                                 & Source \\
    \midrule
	\multicolumn{3}{l}{\textbf{Panel A: First Generation}} \\
	\underline{Costs} && \\
	Medical care per day: 855 USD &  & \citetappendix{Alvearetal2013} \\
	\hspace*{3pt} Year 1: 3.5 days     & 2,992.5 &  \\
	\hspace*{3pt} Year 2: 1.2 days     & 1,026.0 &  \\
	\hspace*{3pt} Year 3: 0.7 days     & 598.5  &  \\
	\hspace*{3pt} Year 4: 0.5 days     & \underline{    427.5    }  &  \\
	PV at birth                        & 4,649.3 &  \\
	Welfare cost of taxation & \underline{2,324.7} & \citetappendix{Garciaetal2020} \\
	\hspace*{0pt}\hfill \textbf{Total} & \textbf{6974.0} & \\
	&&\\
	\underline{Benefits} && \\ 
	Labour market returns to education & 979.7 & \citetappendix{Bharadwajetal2013} \\
	Value of a statistical life & \underline{ 91,760 } & \citetappendix{mardones_riquelme_2018} \\
	\hspace*{0pt}\hfill \textbf{Total} & \textbf{92,739.7} & \\\midrule
	&&\\
	\multicolumn{3}{l}{\textbf{Panel B: Second Generation}} \\
    \underline{Costs} && \\
	Increased probability of VLW & 22.2 & \\
	Labour market returns to education & 86.1 & \citetappendix{BehrmanRosenzweig2018} \\
	Increased infant mortality  & \underline{ 1,573.4 }& \\
	\hspace*{0pt}\hfill \textbf{Total} & \textbf{1,681.7} & \\
	&&\\
	\underline{Benefits} && \\
	Medical care per day: 855 USD &  & \citetappendix{Alvearetal2013} \\
	\hspace*{3pt} Year 2: $\uparrow$0.2 days     & -171.0 &  \\
	\hspace*{3pt} Year 3: 0.5 days     & 427.5 &  \\
	\hspace*{3pt} Year 4: 0.4 days     & 342.0 &  \\
	\hspace*{3pt} Year 5: 0.1 days     & \underline{    85.5    }  &  \\
	\vspace{2mm}
	PV at birth                        & 562.5 &  \\
	PV at mother's birth               & 39.9 &  \\
	Welfare cost of taxation & \underline{ 19.9 } & \citetappendix{Garciaetal2020} \\
	\hspace*{0pt}\hfill \textbf{Total} & \textbf{59.8} & \\
	\bottomrule
    \multicolumn{3}{p{15cm}}{\footnotesize Notes: Each panel displays the costs and benefits in terms of real costs in 2022. Panel A displays the details for generation 1 and panel B for generation 2 considering 0.24 additional births up to 25 years. All costs are presented discounted to the time at birth of the first generation mother. In the case of benefits, these are reported in terms of time accrued, and then discounted to the time of birth of the mother in `PV at mother's birth'.}
    \end{tabular}
\end{table}
\clearpage


\clearpage
\pagenumbering{gobble}
\bibliographystyleappendix{chicago}
\bibliographyappendix{refs}


\end{appendices}
\end{document}
